ENGINEERS' 

SURVEYING  INSTRUMENTS, 


THEIR  CONSTRUCTION,  ADJUSTMENT, 
AND  USE. 


IRA  O.  BAKER,  C.E., 

PROFESSOR    OF    CIVIL    ENGINEERING,    UNIVERSITY     OF     ILLINOIS; 
AUTHOR  OF    A    TREATISE    ON    MASONRY    CONSTRUCTION. 


SECOND  EDITION, 
REVISED    AND    GREATLY    ENLARGED. 


THOUSAND. 


NEW  YORK  : 
JOHN    WILEY    &    SONS 

LONDON  : 
CHAPMAN  &  HALL,  LTD. 

1897. 


COPYRIGHT,  1892, 

BY 
IRA  O.  BAKER. 


75 


ROBKRT   DRUMMOND,    ELECTROTYPER   AND    PRINTER,    NEW   YORK. 


PREFACE. 


THIS  volume  was  prepared  for  the  author's  own 
students,  and  has  been  used  in  his  classes  for  twelve 
years  past,  in  the  form  of  a  blue-print  manuscript  text- 
book. Two  series  of  extracts  were  published  in  two 
engineering  journals  and  afterwards  reprinted  in  book 
form.  As  one  of  these  little  books  had  the  same  title 
as  this  volume,  the  latter  is  gratuitously  called  a  second 
edition. 

The  object  of  this  volume  is  to  acquaint  the  student 
with  the  construction,  adjustment,  and  use  of  surveying 
instruments.  In  no  degree  is  it  intended  for  a  treatise 
on  surveying,  since  there  is  no  lack  of  good  books  on 
the  various  branches  of  that  subject.  It  has  not,  how- 
ever, always  been  possible  to  draw  a  sharp  line  between 
instruction  in  the  use  of  the  instruments,  and  methods  of 
surveying  ;  but,  as  a  rule,  only  such  of  the  latter  have 
been  given  as  are  common  to  all  surveys  made  with  the 
particular  instrument  under  consideration.  In  a  few 
cases,  methods  of  surveying  not  given  in  the  common 
text-books  and  manuals  are  briefly  discussed. 

The  author  began  the  preparation  of  this  volume 
after  becoming  convinced,  by  his  own  observation  as 
well  as  by  that  of  others,  that  students,  as  well  as  many 
practicing  engineers,  would  be  benefited  by  a  more 
thorough  study  of  the  instruments.  The  knowledge  of 
the  instruments  and  the  practice  in  their  use  gained  in 
the  study  of  subjects  in  which  the  instruments  have 

iii 


PREFACE. 


but  subordinate  consideration,  are  not  sufficient  to 
secure  either  economy  of  time  and  effort,  or  maxi- 
mum accuracy.  The  experience  of  the  author,  the  tes- 
timony of  his  students  after  leaving  college,  and  the 
commendations  the  first  edition  received  from  engineers 
in  practice,  have  confirmed  his  opinion  of  the  value  of 
a  detailed  study  of  the  instruments  themselves,  the 
sources  of  error  in  using  them,  the  methods  of  eliminat- 
ing the  errors,  and  the  degree  of  accuracy  attainable. 

In  Appendix  IV  will  be  found  the  problems  which 
the  author  assigned  in  connection  with  a  study  of  the 
text.  These  problems  are  designed  to  familiarize  the 
student  with  the  methods  of  using  the  instruments,  to 
acquaint  him  with  the  qualities  of  good  instruments, 
and  also  to  teach  him  the  degree  of  accuracy  attainable. 
Some  of  the  problems  are  solved  several  times  with 
different  instruments  and  under  different  conditions  as 
to  distance,  weather,  experience,  etc. 

In  the  preparation  of  this  volume  great  care  has  been 
taken  to  present  the  subject  clearly  and  concisely,  and 
to  make  the  book  convenient  for  daily  use  or  ready 
reference.  The  volume  is  divided  into  chapters  and 
articles,  and  it  may  be  helpful  to  the  reader  to  notice 
that  successive  subdivisions  of  the  latter  are  indicated 
by  capital  black-face  side-heads,  by  lower-case  black- 
face side-heads,  by  italic  side-heads,  and  by  simply 
the  serial  section  number.  In  some  cases  the  major 
subdivisions  of  the  sections  are  indicated  by  small  nu- 
merals. The  running  title  at  the  head  of  the  pages 
will  be  of  assistance  in  finding  the  different  parts  of  the 
book.  An  index  at  the  end  of  the  volume  makes  every- 
thing in  the  book  easy  of  access. 

CHAMPAIGN,  ILL.,  Nov.  19,  1892. 


CONTENTS. 


PAGE 

INTRODUCTION.  .  i 


CHAPTER  I.     CHAIN  AND  TAPE. 

Art.  i.  CONSTRUCTION.  Common  Chain.  Steel  Tapes. 

Linen  Tapes .  .  .3 

Art.  2.  TESTING  THE  CHAIN  AND  TAPE.  Standards. 
Testing  the  tape.  Testing  the  chain.  Correcting  for 
error  of  chain 10 

Art.  3.  USING  THE  CHAIN.  How  to  chain.  Chaining  on 
a  slope.  Compensating  vs.  cumulative  errors.  Sources 
of  error.  Limits  of  precision  .  .  .  .  .14 

CHAPTER  II.    TRIPOD,  LEVELING  SCREWS, 
AND   PLUMB-BOB. 

Art.  i.     THE  TRIPOD.     Construction.     Setting  the  tripod    31 
Art.  2.     LEVELING  SCREWS.    Defects  of  ordinary  construc- 
tion         .        .        .        ..        .        .        .        .        .        .33 

Art.  3.     PLUMB-BOB    .        .        •    'S*        •  •        •     35 

CHAPTER  III.     MAGNETIC  COMPASS. 

Art.  i.    CONSTRUCTION.     Prismatic  compass    .        .        .     37 

Art.  2.  TESTS.  The  needle.  Metal  of  compass-box. 
Sights.  Zero  of  vernier  .  .  .  .  .41 

Art.  3.  ADJUSTMENTS.  Levels.  Sights.  Needle.  Cen- 
ter-pin .  .  .  .  .  .  .  .  .  -43 

Art.  4.  USING  THE  COMPASS.  Care.  Practical  hints. 
Sources  of  error.  Limits  of  precision.  Balancing  the 

latitudes  and  departures .46 

v 


VI  CONTENTS. 


CHAPTER  IV.    SOLAR  COMPASS. 

PAGE 

Principle  of  the  Solar  Apparatus 57 

Art.  i.    CONSTRUCTION  OF  THE  SOLAR  COMPASS    .        .    58 
Art.  2.    ADJUSTMENT  OF  THE  SOLAR  COMPASS.        .        .    61 
Art.  3.     USING  THE    SOLAR    COMPASS.    Merits  and  de- 
fects.   History        ,        .        .        .        .        .        .        .62 

CHAPTER  V.     VERNIERS. 

Principles.  To  read  a  vernier.  Practical  hints.  Microm- 
eter .  .  . 64 

CHAPTER    VI.     OPTICAL    PARTS    OF    THE 
TELESCOPE. 

Art.  i.  CONSTRUCTION.  Kinds  of  telescope.  Objective. 

Eye-piece.  Telescope  slide.  Cross  hairs  .  .  .72 

Art.  2.  TESTING  THE  TELESCOPE.  Chromatic  aberration. 
Spherical  aberration.  Definition.  Flatness  of  field. 
Size  of  field.  Aperture  of  objective.  Magnification. 
Illumination .80 

Art.  3.  USING  THE  TELESCOPE.  Adjustment  for  paral- 
lax. Care  of  telescope 88 

CHAPTER  VII.     THE  TRANSIT. 

Art.  i.  CONSTRUCTION.  Graduation,  Verniers.  Centers. 
Levels.  Clamp  and  tangent  screw.  Object-glass  slide. 
Gradienter.  Shifting  plates.  Compass.  Various 
extras 91 

Art.  2.  TESTS  OF  THE  TRANSIT.  Graduation.  Eccen- 
tricity. Magnifying  power  vs.  vernier.  Magnifying 
power  vs.  level  under  telescope.  Magnifying  vs.  plate 
levels.  Parallelism  of  vertical  axes.  Limb  perpendic- 
ular to  axis.  Object-glass  slide  .  .  .  .103 

Art.  3.  ADJUSTMENTS  OF  THE  TRANSIT.  Levels.  Cross 
hairs.  Centering  the  eye-piece.  Standards.  Level 
under  telescope.  Zero  of  the  vertical  circle  .  .109 


CONTENTS.  Vll 


PAGE 

Art.  4.  USING  THE  TRANSIT.  Practical  hints.  Measuring 
angles :  ordinary  method,  by  series,  by  repetition. 
Transit  surveying :  angle  method,  quadrant  method, 
traversing.  Sources  of  error.  Limits  of  precision. 
Care  of  the  transit  .  .  .  .  .  .  .  .115 

CHAPTER  VIII.     SOLAR  TRANSIT. 

Art.  i.    CONSTRUCTION.    Saegmuller's  form.    Other  forms  130 
Art.  2.    ADJUSTMENT.     Polar  axis.     Crosshairs        .        .134 
Art.  3.     USING  THE  SOLAR  TRANSIT.    To  determine  a 
meridian.     To    determine    the  latitude.     Sources  of 
error.     Limits  of  precision.     Solar  transit  in  mine  sur- 
veying  135 

CHAPTER  IX.    PLANE  TABLE. 

Art.  i.     CONSTRUCTION.    Different  forms  :  complete,  light, 

home-made  .  .  .  .  r  .  .  .  .  146 
Art.  2.  TESTS  AND  ADJUSTMENTS.  Sights.  Edge  of  ruler. 

Levels  on  alidade.    Board.   Telescope.    Zero  of  vernier. 

Level  on  telescope  .  .  .  .  -.  .  .  .153 
Art.  3.  USING  THE  PLANE  TABLE.  Methods.  Three-point 

problem.      Two-point    problem.       Sources    of    error. 

Limits  of  precision.     Practical  hints      .        .        .        .155 

CHAPTER  X.     TELEMETERS. 

Art.  i.  STADIA.  Principles.  Placing  the  hairs.  The 
rod.  Formula  for  horizontal  line  of  sight  and  vertical 
rod.  Position  of  rod  for  inclined  line  of  sight.  For- 
mulas for  inclined  line  of  sight  and  vertical  rod.  Re- 
ducing field  notes :  diagrams,  tables.  Sources  of  error. 
Limits  of  precision.  Practical  hints  .  .  .  .  173 

Art.  2.  GRADIENTER.  As  a  leveling  instrument.  As  a 
telemeter.  Limits  of  precision.  Vertical  circle  as  a 
gradienter .  .  209 

Art.  3.     VARIOUS  TELEMETERS    .  ....  216 


Vlll  CONTENTS. 


CHAPTER  XL     SPIRIT  LEVELS. 

PAGE 

Art.  i.  CONSTRUCTION  OF  THE  INSTRUMENT.  Qualities 

desired.  Classification .218 

Art.  2.  CONSTRUCTION  OF  THE  ROD.  Target  rods.  Self- 
reading  rods  .  .  .  .  ...  .  .  227 

Art.  3.  TESTING  THE  LEVEL.  Bubble  tube.  Sensitive- 
ness vs.  magnifying  power.  Telescope  slide.  Rings 
and  wyes  of  wye  level  .  ...  .  .  .  236 

Art.  4.  ADJUSTMENTS  OF  WYE  LEVEL.  Level  tube. 
Cross  hairs.  Centering  eye-piece.  Wyes.  Test  level  241 

Art.  5.  ADJUSTMENTS  OF  DUMPY  LEVEL.  Collimation. 
Wyes .  .  . .  .247 

Art.  6.    ADJUSTMENTS  OF  LEVEL  OF  PRECISION     .        .  249 

Art.  7.  USING  THE  LEVEL.  Differential  leveling.  Profile 
leveling.  Precise  leveling.  Sources  of  error.  Limits 
of  precision.  Practical  hints.  Care  of  the  level  .  .  252 

CHAPTER  XII.     BAROMETERS. 

Art.  i.  MERCURIAL  BAROMETER.  Construction.  Clean- 
ing. Filling.  Reading.  Transporting  .  .  .  292 

Art.  2.  ANEROID  BAROMETER.  Common  or  Vidi's.  Gold- 
schmid's  .  ...  .  .  .  ...  .  >  .  301 

Art.  3.  THE  PRACTICE.  Sources  of  error  :  gradient,  tem- 
perature, humidity,  instrumental,  observational.  Limits 
of  precision.  Methods  of  observing:  single  observa- 
tion, simultaneous  observations,  observations  at  se- 
lected times,  Williamson's  method,  Whitney's,  Planta- 
mour's,  Riihlmann's,  Gilbert's  .  .  .  .  308 

Art.  4.  BAROMETRIC  FORMULAS.  A.  Statical  Formulas. 
Assumptions.  Fundamental  Relations.  Typical 
formulas  :  Laplace's,  Babinet's,  Bailey's,  Plantamour's, 
Williamson's.  Altitudes  cales  of  aneroids.  B.  Dynam- 
ical Formulas.  Ferrel's,  Gilbert's,  Weilenmann's.  .  327 

APPENDIX    I.    ELIMINATION    OF    LOCAL 
ATTRACTION  IN  SURVEYING  WITH 

THE  MAGNETIC  COMPASS. 
Art.  i.     MINE  SURVEYS        .        .       .       ...        .        .349 

Art.  2.     LAND  SURVEYS        .   •..."     .       .  .        .  356 


CONTENTS.  IX 


APPENDIX    II.     AREA  BY  TRAVERSING  WIT 
TRANSIT. 

PAGE 

Taking  the  notes.     Computing  the  area      ....  360 

APPENDIX  III.     PROBABLE  ERROR. 
Principles.     Formulas.     Examples        .        .        .  .  366 

APPENDIX  IV.    PROBLEMS. 

Chain.  Magnetic  Compass.   Transit.    Solar  Transit.   Plane 
Table.     Stadia.    Level.     Barometer     ....  373 


ENGINEERS'  SURVEYING  INSTRUMENTS. 


INTRODUCTION. 

THE  importance  to  the  engineer  of  a  knowledge  of 
the  best  forms  of  construction  of  his  instruments  and 
of  a  thorough  understanding  of  the  principles  which 
should  govern  their  adjustments,  is  self-evident ;  and  it 
is  no  less  evident  that  he  should  be  expert  in  handling 
his  instruments.  The  general  plan  of  this  discussion 
will  be  (i)  to  call  attention  to  the  principal  points  in  the 
common  forms  of  construction  of  each  instrument,  not- 
ing the  advantages  and  disadvantages  of  each  ;  (2)  to 
consider  certain  relations  which  the  parts  of  a  perfect 
instrument  should  bear  to  each  other,  and  which  are 
supposed  to  be  adjusted  by  the  maker  once  for  all,  ex- 
plaining how  the  engineer  may  test  them  for  himself 
although  he  may  not  be  able  to  correct  them  ;  (3)  to 
explain  the  method  of  making  the  several  adjustments  ; 
(4)  to  describe  the  method  of  using  the  instrument 
which  will  secure  the  most  accurate  results  in  the  short- 
est and  easiest  way,  and  to  note  the  various  sources  of 
error  to  which  the  work  is  liable,  and  also  to  give  data 
to  show  the  degree  of  accuracy  attainable  ;  and  (5)  to 
add  a  few  hints  on  the  proper  care  of  the  instrument. 
It  will  be  unnecessary  to  describe  minutely  the  different 


INTRODUCTION. 


parts  of  the  several  instruments  and  the  objects  of  each, 
except  as  is  requisite  in  carrying  out  the  above  outline  ; 
for  more  can  be  learned  by  a  moment's  inspection  of 
the  instrument  than  from  any  printed  description.  No 
attempt  will  be  made  to  describe  any  of  the  special 
devices  proposed  by  the  different  instrument-makers, 
the  desire  being  to  state  the  general  principles  which 
will  enable  the  reader  to  judge  of  the  merits  of  any 
modification  of  the  ordinary  forms. 


CHAPTER  I. 
CHAIN  AND  TAPE. 

ART.  1.     CONSTRUCTION. 

1.  COMMON  CHAIN.  The  ordinary  chain  consists  of 
one  hundred  pieces  of  wire,  called  links,  bent  into  rings 
at  the  ends  and  connected  together  by  two  (sometimes 
three)  rings.  In  using  the  chain  a  "  link  "  includes  a 
ring  at  each  end.  Iron  chains  are  made  of  two  sizes  of 
wire,  Nos.  8  and  10,  the  former  being  about  five  thirty- 
seconds  of  an  inch  in  diameter  and  the  latter  nearly 
one  eighth  of  an  inch.  Steel  chains  are  made  of  No. 
10  or  12  wire,  the  latter  being  about  seven  sixty-fourths 
of  an  inch  in  diameter.  The  best  chains  are  made  of 
No.  12  tempered-steel  wire.  All  joints  in  the  rings 
and  links  should  be  brazed  to  prevent  their  opening, 
and  the  consequent  lengthening  of  the  chain.  The 
chain  is  divided  decimally  by  brass  tags  numbered  from 
each  end  to  the  middle. 

The  surveyor's  chain  (Gunter's  chain)  is  66  feet  long, 
each  "link"  being  7.92  inches.  It  is  used  only  in  find- 
ing the  area  of  land  where  fhe  acre  is  the  unit  of 
measure,  and  is  much  less  frequently  used  for  this  pur- 
pose now  than  formerly.  The  66-foot  chain  is  used  on 
all  the  U.  S.  public-land  surveys  ;  and  in  all  deeds  of 
conveyance  and  other  documents,  when  the  word  chain 
is  used,  it  is  Gunter's  chain  that  is  meant. 


CHAIN    AND    TAPE.  [CHAP. 


An  engineer's  chain  is  100  feet  long,  each  "link"  be- 
ing i  foot.  This  chain  is  used  in  surveying  railroads, 
canals,  and  where  extensive  line  surveys  are  being  con- 
ducted. It  is  not  infrequently  employed  in  finding 
areas  in  acres.  It  is  preferred  to  the  surveyor's  chain 
on  account  of  its  greater  length,  which  enables  one  to 
work  more  rapidly  and  more  accurately. 

When  the  chain  is  folded  up  the  links  should  not  be 
parallel  to  each  other,  but  should  be  crossed  in  such  a 
manner  as  to  touch  each  other  in  the  middle,  thus  pre- 
venting the  bending  of  the  links  in  tying  up  the  chain. 
See  Fig.  i. 


FIG.  i. 

2.  Advantages  and  Defects  of  the  Common  Chain.  The 
principal  advantages  of  the  chain  are  its  flexibility  and 
the  ease  with  which  its  subdivisions  are  distinguished. 

The  ordinary  chain  is  defective  on  account  of  (i)  un- 
avoidable wearing  of  the  numerous  points  of  contact,  (2) 
opening  of  the  rings,  (3)  lengthening  by  stretching  and 
by  flattening  of  the  rings,  (4)  shortening  by  mud,  ice,  or 
grass  getting  into  the  joints,  (5)  varying  in  length  with 


ART.    l]  CONSTRUCTION. 


temperature,  and  (6)  kinking.  Some  of  these  defects 
are  so  small  as  to  be  appreciable  only  in  very  careful 
work  ;  and  some  exist  even  in  the  most  elaborate  ap- 
paratus that  can  be  devised.  If  each  of  the  six  hundred 
points  of  contact  of  the  common  loo-link  chain  wears 
only  one  hundredth  of  an  inch,  the  length  of  the  chain 
is  increased  6  inches.  Mud  and  ice  in  the  joints  have  a 
still  greater  effect  in  the  opposite  direction. 

3.  STEEL  TAPES.  Steel  ribbons  are  now  made  of  any 
length  up  to  1,000  feet  without  joint  or  splice  from 
end  to  end.  In  width  these  ribbons  vary  from  an 
eighth  of  an  inch  to  a  half  inch,  and  in  thickness  from 
one  hundredth  to  four  hundredths  of  an  inch.  For  gen- 
eral work  a  tape  about  a  quarter  of  an  inch  wide  and 
two  hundredths  of  an  inch  thick,  blued  and  polished  or 
nickel-plated,  is  generally  preferred.  The  wider  and 
thinner  tapes  are  nearly  useless  in  general  field-work, 
owing  to  the  ease  with  which  they  are  broken. 

The  divisions  of  the  tape  are  marked  in  several  ways, 
viz.:  (i)  the  numbers  and  graduation  marks  are  stamped 
in  a  piece  of  brass  which  is  soldered  on  the  tape  ;  (2) 
the  marks  are  countersunk  in  a  lump  of  solder  attached 
to  the  tape  for  that  purpose  ;  (3)  the  foot  divisions  are 
marked  by  single  rivets  and  the  lo-foot  divisions  by  a 
brass  burr  riveted  on,  the  10,  20,  etc.,  marks  being  distin- 
guished either  by  numbers  countersunk  in  the  brass 
burr  or  by  i,  2,  etc.,  small  rivets  ;  or  (4)  the  face  of 
the  tape  is  etched  with  acid  so  that  the  division  marks 
and  figures  stand  out  in  relief  while  the  etched  surface 
appears  dull.  No  method  of  indicating  the  graduation 
is  entirely  satisfactory,  but  the  first  and  second  are  prob- 
ably best.  The  first  is  probably  more  durable  than  the 
second,  although  the  graduation  marks  are  liable  to 
wear  or  tear  off,  particularly  on  gravel  or  stony  ground. 
In  either  case  there  should  be  two  series  of  numbers, 
one  counting  from  each  end.  It  is  not  desirable  to  have 


CHAIN    AND    TAPE.  [CHAP.   I 


the  graduation  indicated  by  rivets,  since  the  tape  is 
weakened  by  the  rivet-hole,  and  also  by  the  accumula- 
tion of  moisture  under  the  rivet  head.  One  advantage 
of  this  method  is  that  the  graduation  can  be  read  from 
either  side  of  the  tape.  The  first  and  second  methods 
are  sometimes  employed  upon  the  same  tape.  The  last 
method  is  employed  only  for  pocket  tapes  graduated  to 
feet  and  inches,  owing  to  the  liability  of  weak  places 
from  over-etching. 

Steel  tapes  graduated  by  any  of  the  first  three 
methods  mentioned  above  are  frequently  called  band 
chains  and  sometimes  chain  tapes,  to  distinguish  them 
from  tapes  graduated  by  the  fourth  method, /.<?.,  gradu- 
ated fully  throughout. 

Steel  tapes  can  be  had  graduated  to  feet  and  inches, 
to  feet  and  tenths,  to  links,  or  metrically.  The  wide 
thin  tapes  graduated  to  feet  and  inches  by  etching  are 
very  convenient  in  city  surveying,  or  where  accurate 
measurement  of  irregular  distances  is  frequently  re- 
quired. The  tape  ordinarily  employed  in  general  field- 
work  is  a  narrow  ribbon  graduated  to  feet,  with  each  end 
foot  to  tenths.  This  form  of  tape  would  be  greatly  im- 
proved if,  instead  of  subdividing  the  first  and  last  foot 
of  the  hundred,  an  additional  foot  were  added  at  each 
end,  and  so  divided.  When  the  end  foot  of  the  hundred 
is  subdivided,  if  one  desires  to  lay  off,  say,  28.3  feet,  one 
chain-man  must  hold  the  tape  at  29  feet  and  the  other 
at  7  tenths  from  the  end  of  the  graduation,  which  oper- 
ation produces  mental  confusion  and  is  apt  to  cause 
error,  since  neither  chain-man  holds  the  tape  at  the 
number  to  be  laid  off.  With  an  extra  foot  at  the  end  of 
the  tape,  one  chain-man  holds  the  tape  at  the  whole 
number  of  feet  to  be  laid  off,  while  the  other  chain-man 
holds  at  the  number  corresponding  to  the  fraction  to  be 
laid  off. 

Frequently  the   end    graduation   of   a    steel    tape   is 


ART.    l]  CONSTRUCTION.  7 

simply  a  line  across  the  face  of  the  tape  or  a  rivet  in  the 
middle  of  its  width,  in  which  case  it  is  not  possible  to 
stick  the  pin  quickly  or  exactly  opposite  the  mark. 
The  end  graduation  mark  should  be  indicated  by  the 
square  shoulder  of  a  piece  of  brass  soldered  or  riveted 
to  the  tape  for  that  purpose.  These  shoulders,  if  put 
on  at  all,  generally  face  opposite  ends  of  the  tape. 
They  should  both  face  the  same  end  (preferably  the  rear 
end)  of  the  tape,  as  shown  in  Fig.  2  ;  in  which  case  the 


FIG.  2. 

same  side  of  the  pin  can  be  used  by  both  the  fore  and 
hind  chain-man. 

4.  Reels.  For  convenience  of  transportation  steel 
tapes  are  wound  up  either  on  an  open  reel  or  in  a 
tight  case,  the  narrower  thicker  ones  on  the  former  and 
the  wider  thinner  ones  in  the  latter.  In  one  respect  the 
open  reel  is  better  than  the  tight  case,  since  with  the 
former  the  tape  has  a  better  chance  to  dry  off.  It  is 
desirable  that  the  reel  should  be  strong,  durable,  and 
convenient,  and  at  the  same  time  be  light  and  of  such 
a  form  as  to  be  carried  in  the  pocket  when  the  tape  is 
in  use.  No  reel  that  the  writer  has  seen  fulfills  this 
condition  fairly  well.  Generally  the  axis  about  which 
the  tape  is  wound  is  so  small  as  to  break  the  tape  near 
the  end. 

Most  of  the  reels  require  that  one  handle  be  re- 
moved before  beginning  to  wind  up  the  tape,  which 
necessitates  the  handles  being  easily  detached  without 
liability  to  come  off  when  in  use.  Most  of  the  handles 
on  the  market  are  deficient  in  one  or  the  other  of  these 


8  CHAIN    AND    TAPE.  [CHAP.  I 

respects.  On  the  whole,  it  is  probably  best  to  wind  the 
tape  up  in  the  hands  in  the  form  of  a  figure  eight  (8) 
about  two  feet  long,  and  tie  with  a  string  at  the 
center. 

5.  Advantages  and  Disadvantages  of  Steel  Tapes.     Steel 
tapes   have   superseded  chains  for   nearly  all  kinds   of 
work  for  the  following  reasons:    (i)    Tapes  are  lighter 
than  chains  of  equal  strength.     (2)  Tapes  being  smooth 
and  having  no  projections  are  easier  to  drag.     (3)  Tapes 
do  not  alter  their  length  by  wear. 

The  disadvantages  are:  (i)  The  steel  tape  will  not 
stand  as  rough  handling  as  the  chain.  (2)  It  can  not 
be  repaired  in  the  field  when  broken,  while  the  chain 
can  lose  one  or  more  of  its  links  and  still  be  of  service. 
(3)  The  graduation  is  not  as  legible  as  that  of  the 
chain,  and  becomes  obliterated  with  use. 

Notwithstanding  its  defects,  a  steel  tape  is  vastly 
better  than  a  chain  for  nearly  all  kinds  of  work. 

6.  Wood  Rods  vs.  Steel  Tapes.     Before  the  introduc- 
tion  of  steel    tapes   short    wood   rods   were    employed 
when  great  accuracy  was  necessary.      They  were  free 
from  some  of  the  imperfections  of  the  common  chain, 
but   were  defective   in    other  and  generally  more   im- 
portant respects.     Short  rods  should  be  used  only  with 
the  accurate  alignment  and  elaborate  means  of  secur- 
ing delicacy  of  contact  employed  in  geodetic  measure- 
ments ;  and  recent  experience*  seems  to  show  that  in 
measuring  geodetic  base  lines,  work  can  be  done  with 
greater  speed  and  accuracy  with  steel  tapes  than  with 
the    most   elaborate  and  expensive  base  apparatus  yet 
devised. 


*  See  Annual  Report  of  the  Missouri  River  Commission  for  1886,— Execu- 
tive Document  No.  28,  49th  Congress,  2<d  Session, — pp.  31-35.  For  method  of 
accurate  measurement  with  two  steel  tapes  or  wires  used  as  a  metallic  ther- 
mometer, see  Annual  Report  of  the  Chief  of  Engineers,  U.  S.  A.,  for  1890, 
pp.  1838-45. 


ART.    l]  CONSTRUCTION. 


7.  Steel  Wires.  A  steel  wire  makes  a  good  substitute 
for  a  chain.  Fig.  3  shows  a  method  of  indicating  the 
end  of  the  distance  to  be  laid  off.  ab  is  a  small  rod 
with  a  hole  through  it,  into  which  the  wire  is  fastened; 
c  is  a  V-shaped  groove  around  the  rod.  The  pin  may 
be  set  in  the  groove  c  without  any  danger  of  the  back 
pin's  being  displaced  by  the  jerking  of  the  wire  by  the 
fore  chain-man  in  getting  it  into  line;  or  the  distance 
may  end  at  a  or  £,  in  which  case  it  can  be  more  accu- 
rately marked  with  a  knife  in  a  board.  The  ends  a 


FIG.  3. 

and  b  should  be  exactly  at  right  angles  to  the  direction 
of  the  length  of  a  b.  A  handle  can  be  formed  by  pass- 
ing the  wire  through  a  piece  of  wood  to  protect  the 
hand,  and  then  bending  the  wire  into  a  loop.  Unfor- 
tunately fractional  parts  of  the  unit  are  not  easily  ob- 
tained; but  in  many  cases  only  units  are  required. 

8.  LINEN     TAPES.     Although  accurate  work  can  not 
be  done  with  linen  tapes,  they  are  useful  in  many  kinds 
of  work,  as,  for  example,  in   cross-sectioning  for    rail- 
road   earthwork.       A    linen     tape    contracts     in    wet 
weather   and    expands    in    dry,    and    it   can    easily    be 
permanently  elongated  by  overstraining. 

9.  Metallic  Tapes  are  a  species  of   linen    tapes  hav- 
ing fine  brass  wires  woven  through  their  entire  length. 
Metallic  tapes  do   not  vary  in   length  with   changes  in 
the  hygrometric  state  of  the  atmosphere  as  much  as 
linen  tapes,  and  are  not  so  easily  elongated  by  over- 
straining. 


10  CHAIN    AND    TAPE.  [CHAP.  I 


ART.  2.     TESTING  THE  CHAIN  AND  TAPE. 

10.  STANDARDS.     Each  instrument-maker  claims  that 
the  chains  and  tapes  which  he  sends  out  are  true  U.  S. 
standard;  but  it  is  certain  that  chains  and  tapes  from 
different  makers,  purporting  to  be  of  the  same  standard, 
differ  in  length.    Even  if  correct  in  the  beginning,  chains 
wear  and  tapes  break;  and  therefore  neither  are  likely  to 
be  of  the  same  length  after  being  used  some  time.    Hence 
every  engineer  should  have  some  means  of  testing  the 
length  of  his  chain  or  tape.     The  simplest  way  to  get  a 
reliable  standard  is  to  send  a  new  tape  to  the  Superin- 
tendent of  the  U.  S.  Coast  and  Geodetic  Survey,  Wash- 
ington,  D.  C.,  who  will  compare  it  with  the  standard 
and   place  the  government  stamp  on  it   to  show  its  de- 
gree of  accuracy.     For  a  degree  of  accuracy  sufficient 
for  ordinary  engineering  a  fee  of  50  cents  is  charged; 
but  when  extreme  accuracy  is  desired  the  fee  is  higher. 

The  tape  so  compared  should  be  reserved  for  testing 
other  tapes,  or  the  standard  distance  should  be  carefully 
laid  off  on  the  floor  of  a  building,  or  on  a  stone  water- 
table,  or  on  two  stone  or  iron  posts  firmly  set  in  the 
ground  for  that  purpose,  or  in  any  way  that  shall  per- 
manently preserve  the  exact  distance.  When  laying  off 
the  standard  distance  the  tape  should  be  supported 
throughout  its  entire  length;  should  be  perfectly 
straight,  both  horizontally  and  vertically,  and  should  be 
stretched  with  the  same  tension  as  that  employed  in 
testing  it  at  Washington.  In  laying  off  this  distance 
the  temperature  of  the  tape  should  be  carefully  noted, 
and  if  it  is  not  the  same  as  that  at  which  the  tape  is  a 
standard,  a  correction  (see  §  19,  paragraph  e)  should  be 
applied  before  making  the  permanent  mark. 

11.  It  is  sometimes  held  that  since  the  chain  when  in 
use  is  seldom    if   ever  stretched    perfectly  straight,  it 


ART.  2]  TESTING    THE   CHAIN    AND    TAPE.  II 

should  be  made  a  little  longer  than  the  standard  so  that 
the  full  length  of  the  standard  may  be  laid  off  each 
time.  The  instructions  issued  by  the  U.  S.  Land  Office 
to  Surveyors-General  states  *  that  "  the  66-foot  chain 
shall  be  66.06  ft.,"  for  the  above  reason.  The  French 
have,  or  at  least  had,f  a  similar  practice,  the  addition 
being  from  r  in  1,000  to  i  in  2,000.  In  all  the  arts  de- 
pending in  any  way  upon  accuracy  of  measurements 
there  is  great  confusion  on  account  of  differences  be- 
tween so-called  standards.  In  some  cases,  particularly 
that  of  iron  work,  this  diversity  arose  in  a  manner  sim- 
ilar to  that  referred  to  above.  The  chain  should  be 
exactly  as  long  as  the  standard. 

12.  Notice  that  in  many  cases  the  standard  by  which 
the  engineer  is  to  test  his  chain  or  tape  is  not  an  abso- 
lute one.  For  example,  the  law  under  which  the  U.  S. 
public  lands  were  surveyed  says  "  all  the  corners  marked 
in  the  surveys  returned  by  the  Surveyor-General  shall 
be  established  as  the  proper  corners,"  etc.,  and  "  the 
length  of  such  lines  as  returned  shall  be  held  and  considered 
as  the  true  length  thereof."  This  law  establishes  a  stand- 
ard of  measure  between  every  pair  of  adjacent  corners 
of  the  government  survey,  and  this  standard  is  the  only 
one  that  can  legally  be  used  in  measuring  that  line.  As- 
sume, for  example,  that  a  surveyor  being  called  upon  to 
establish  the  corner  at  the  middle  of  the  west  side  of 
sec.  i,  measures  the  west  side  of  that  section  and  finds 
it  to  be  79.26  chains  by  his  chain,  while  the  recorded 
distance  is  79.83  chains.  Then,  since  he  must  consider 
the  recorded  distance  as  the  true  distance,  the  true 
length  of  his  chain  is  79.83  -=-  79.26  =  1.007  times  the 
standard.  Since  the  surveyor  is  to  establish  a  corner  40 
chains  north  of  the  south-west  corner  of  sec.  i,  the  dis- 


*  In  the  Instructions  for  1880,  for  the  first  time. 
t  Gillespie's  Land  Surveying,  page  18,  foot-note. 


12  CHAIN    AND    TAPE.  [CHAP.  I 

tance  as  measured  by  his  chain  is  40  -f-  1.007  =  39.72. 
A  similar  principle  applies  in  city  surveying  when  the 
land  is  described  as  being  a  certain  lot  in  a  particular 
block  of  a  recorded  plat. 

If  the  land  to  be  surveyed  is  described  by  metes  and 
bounds,  then  it  is  important  that  the  surveyor  shall  lay 
off  true  standard  distances. 

13.  TESTING   THE    TAPE.     In   comparing  a  tape  with 
the  standard  distance  laid  off  as  above,  the  tape  should 
be  under  the  same  conditions  as  to  temperature,  tension, 
etc.,  that  it  is  to  have  when  in  use.     The  men  who  are 
to  use  the  tape  should  test  it  that  they  may  better  under- 
stand the  proper  tension  required. 

14.  TESTING  THE  CHAIN.     The   same  precautions  are 
to  be  observed  in  testing  the  chain  as  for  the  tape.     The 
length  of  the  chain  should  be  considered  as  the  distance 
from  the  inside  of  one  handle  to  the  outside  of  the  other 
(see  §  16).     In  any  case,  the  points  considered  as  the 
ends  of  the  chain  depend  upon  the  manner  of  setting 
the  pins. 

The  custom  of  taking  up  the  wear  of  the  chain  by  a 
screw  at  one  end  is  wrong  in  principle,  although  it  does 
not  produce  much  error  in  practice.  The  increased 
length  is  produced  by  wear  at  every  joint  ;  taking  it  up 
at  the  ends  destroys  the  equality  of  the  scale  of  equal 
parts,  and  when  only  a  fractional  part  of  the  chain  is 
used  an  error  is  produced.  The  better  method  is  to 
compare  frequently  the  chain  with  the  standard,  and 
apply  a  correction  to  the  measured  distance  or  computed 
area. 

15.  CORRECTING  FOR  ERROR  OF  CHAIN.    Owing  to  wear 

it  frequently  happens  that  the  chain  is  not  of  stand- 
ard length,  and  in  using  a  tape  in  making  re-surveys 
the  tape  often  does  not  agree  with  the  unit  used  in  lay- 
ing off  the  line  which  is  being  re-measured.  Conse- 
quently it  is  often  necessary  to  correct  a  measured  dis- 


ART.  2]  TESTING    THE    CHAIN    AND    TAPE.  13 

tance  for  error  of  chain.  If  in  testing  the  chain  by  the 
method  of  §  10  or  §  12  it  is  found  to  be  four  tenths  of  a 
foot  (or  link)  too  long,  then  the  chain  is  1.004  times  the 
true  standard  ;  and  if  the  distance  as  measured  is  273.8 
ft.,  the  true  distance  is  273.8  X  1.004  =  274-9  ft. 

If  the  area  is  required,  make  the  computations  as 
though  the  chain  were  correct  ;  and  then  the  true  area 
is  equal  to  the  computed  area  multiplied  by  the  square 
of  the  length  of  the  chain  in  terms  of  the  standard.  For 
example,  assume  that  the  computed  area  is  10.875  acres, 
and  that  the  chain  is  1.002  times  the  standard.  Then 
the  true  area  is  10.875  X  (i.oo2)a=  10.919  acres.  Notice 
that  the  length  of  the  chain  can  be  expressed  thus  :  i 
+  .002  ;  and  that  the  square  of  that  quantity  can  be  ex- 
pressed thus  :  (i  +  .oo2)a  =1-1-2  (.002)  +  (.oo2)2.  Since 
the  square  of  .002  is  very  small,  it  may  be  omitted,  and 
then  (i  +  .002)*  =  1  +  2  (.002)  nearly.  Hence  the  cor- 
rection to  the  area  is  equal  to  the  computed  area  multi- 
plied by  twice  the  correction  of  the  chain.  For  illustra- 
tion, the  true  area  for  the  above  example  is  10.875  ~i~ 
(10.875  X  2  X  0.002)  =  10.918  acres.  In  this  example 
the  results  by  the  two  methods  differ  by  less  than  one 
thousandth  of  an  acre,  and  in  this  class  of  problems  the 
error  will  always  be  inconsiderable.  If  the  chain  is,  say, 
.002  too  short,  then  its  length  is  expressed  by  the  for- 
mula i  —  .002,  and  the  correction  to  the  area  is  found 
as  above,  except  that  it  must  be  subtracted. 

The  student  is  cautioned  to  remember  that  if  the  chain 
is  too  long  the  distance  and  the  area  are  too  small  ;  and, 
vice  versa,  if  the  chain  is  too  short  the  distance  and  the 
area  are  too  large.  Very  frequently  errors  are  made  by 
applying  this  correction  in  the  wrong  way. 


14  CHAIN    AND    TAPE.  [CHAP.  I 


ART.  3.    USING  THE  CHAIN. 

16.  HOW  TO  CHAIN.  The  following  is  the  general 
method" oTsgrocedure  in  chaining,  but  is  frequently 
modified  as  circumstances  require.  To  measure  a  dis- 
tance with  a  chain,  two  men  are  required,  a  fore  chain- 
man  and  a  hind  chain-man.  The  hind  chain-man  has  the 
more  responsible  position.  They  should  be  provided 
with  eleven  marking  pins. 

Supposing  the  chain  to  be  tied  up,  the  fore  chain- 
man  throws  it  out  in  the  direction  opposite  to  that  in 
which  the  chaining  is  to  be  done,  gives  the  hind  chain- 
man  a  pin,  takes  nine  in  his  left  hand  and  the  end  of 
the  chain  and  one  pin  in  his  right  hand,  and  draws  the 
chain  in  the  direction  of  the  line.  The  hind  chain-man 
examines  the  chain  as  it  passes,  to  see  that  there  are 
no  kinks  or  bent  links;  or,  if  a  tape  is  used,  to  see  that 
there  are  no  loops  in  it. 

When  the  fore  chain-man  has  gone  the  proper  dis- 
tance, he  stops,  rests  his  right  elbow  on  his  right  knee, 
and  extends  his  right  hand,  in  which  he  holds  the 
handle  and  the  pin,*  as  far  as  possible  from  his  body, 
so  that  the  hind  chain-man  may  have  an  unobstructed 
view  of  the  pin  and  the  farther  end  of  the  line.  The 
fore  chain-man  is  to  keep  the  chain  straight  and  taut, 
and  obey  the  signals  of  the  hind  chain-man. 

The  hind  chain-man  places  his  end  of  the  chain  at 
the  point  of  beginning,  and,  by  placing  himself  behind 
the  point,  with  a  motion  of  his  arm  directs  the  fore 
chain-man  where  to  place  his  pin.  For  example,  if  the 
pin  ought  to  be  moved  a  considerable  distance  to  the 

*  On  a  steel  tape  the  handle  extends  beyond  the  end  graduation,  and  hence 
the  fore  chain-man  should  grasp  the  handle  of  the  tape  in  his  left  hand,  rest 
his  left  elbow  on  his  left  knee,  and  hold  the  pin  in  his  right  hand,  instead  of  as 
described  above  for  the  chain. 


ART.  3]  USING    THE    CHAIN.  15 

right,  the  right  arm  is  held  far  out  from  that  side  of  his 
body;  if  it  should  be  moved  only  a  little,  the  arm  is 
held  nearly  vertical.  The  signal  thus  indicates  both 
the  direction  and  the  amount  of  motion  required.  As 
the  pin  approaches  the  proper  position,  the  arm  comes 
more  nearly  vertical;  and  when  the  pin  is  at  the  proper 
place,  the  hind  chain-man  calls  out  "  stick."  The  fore 
chain-man  then  brings  his  left  hand  to  bear  on  the  top 
of  the  pin,  and  forces  it  vertically  into  the  ground.* 
After  the  pin  is  set,  he  should  test  it  to  see  that  the 
pin,  at  the  surface  of  the  ground,  is  just  in  contact 
with  the  front  face  of  the  handle.  When  the  position  of 
the  pin  is  satisfactory  to  the  fore  chain-man,  he  calls  out 
"  stuck."  At  this  signal  the  hind  chain-man  loosens  his 
end  of  the  chain,  and  both  move  forward  the  length  of 
the  chain. 

The  leader  should  keep  his  eye  steadily  on  the  far- 
ther end  of  the  line,  so  that  he  may  keep  near  the  line. 
When  the  follower  reaches  the  pin  already  set,  he  calls 
"  halt,"  and  the  leader  prepares  to  set  a  pin.  After 
the  fore  chain-man  has  placed  his  pin  in  line,  the  fol- 
lower drops  his  end  of  the  chain  over  the  pin,f  at  the 
same  time  placing  his  hand  on  the  pin  to  hold  it  firm,J 
and  calls  out  "  stick."  At  the  reply  "  stuck,"  he  re- 
moves his  pin  and  the  work  proceeds  as  before. 

When  the  leader  has  set  his  last  pin,  he  calls  "  tally; " 
and  the  hind  chain-man  comes  up,  and  gives  the  fore 


*  When  the  tape  is  used,  the  end  of  it  is  held  in  the  left  hand  and  the  pin 
is  forced  into  the  ground  with  the  right  hand. 

t  Notice  that  both  men  measure  to  the  same  side  of  the  pin. 

%  The  liability  of  the  back  pin's  being  pulled  over  in  this  operation  is  an 
objection  to  this  method  ;  but  it  can  be  eliminated  by  having  the  fore  chain- 
man  set  the  pin  on  the  inside  of  his  handle,  while  the  back  chain-man  simply 
brings  the  outside  of  his  handle  against  the  front  side  of  the  pin.  However^ 
it  is  very  difficult  for  the  fore  chain-man  to  set  the  pin  on  the  inside  of  the 
handle.  Notice  that  with  a  properly  made  steel  tape  (see  last  paragraph  uf 
§  3)  neither  of  these  difficulties  will  occur. 


l6  CHAIN    AND    TAPE.  [CHAP.  I 

chain-man  the  ten  pins  which  he  has,  both  men  count- 
ing them  to  be  sure  that  none  have  been  lost.  The 
follower  then  makes  note  of  the  tally,  and  the  work 
proceeds  as  before. 

When  the  leader  reaches  the  end  of  the  line,  he 
stops,  holds  his  end  of  the  chain  against  it  and  calls 
"  stuck."  The  follower  then  comes  forward  and  counts 
the  distance  beyond  the  last  pin,  being  careful  to 
notice  on  which  side  of  the  middle  the  pin  is.  Each 
tally  represents  ten  chains,  each  pin  held  by  the  fol- 
lower (not  including  the  one  in  the  ground)  represents  a 
chain,  and  the  feet  just  counted  make  up  the  total  dis- 
tance. Notice  that  the  pin  last  set  is  not  counted.  It 
should  always  remain  in  the  ground  until  the  distance 
is  recorded. 

17.  Chaining  on  a  Slope.  In  nearly  all  cases,  it  is 
the  horizontal  distance  which  is  required.  Therefore 
it  is  necessary  to  determine  the  flattest  slope  that  must 
be  taken  into  account.  The  difference  between  the  dis- 
tance measured  on  the  slope  and  the  true  horizontal 
distance  is  given  very  nearly*  by  the  formula 


in  which  d  is  the  difference  sought,  and  /  the  rise  of  the 
slope  in  any  distance  b.  The  quantities  a7,/,  and  b  all 
must  be  in  the  same  unit.  For  example,  if  the  slope 

*  To  determine  the  difference  between  the  base  and  perpendicular  of  a  right- 
angle  triangle,  represent  the  base  by  £,  the  hypotenuse  by  ^,  and  the  perpen- 
dicular by  p.  Then  we  have 


b  =  \h*-p*=(fr-p*)*=  h  -,  nearly. 
Similarly, 

h  =  !/$»+/»  =  (i>*  +/2}*  =  b  +  £,  nearly. 
Therefore  we  may  derive  this  simple  rule  :  The  difference  between  the  base 


ART.  3] 


USING    THE    CHAIN. 


rises   2   feet   in   100  feet,  then  d=~- ,=  ~^—  =  0.02   ft. 

zb      200 

Hence,  if  the  slope  is  2  in  100,  measuring  on  the  slope 
causes  an  error  of  i  in  5,000.  Disregarding  a  slope  of 
3  in  100  causes  an  error  of  nearly  i  in  2,000,  and  disre- 
garding a  slope  of  4  in  100  causes  an  error  of  i  in  1,200. 

If  the  slope  is  too  great  to  be  disregarded,  the  ques- 
tion then  arises  as  to  the  best  method  of  eliminating 
the  error.  Either  of  two  methods  may  be  employed  : 
(i)  one  end  of  the  chain  may  be  raised  to  a  level  with 
the  other,  or  (2)  the  measurement  may  be  made  on  the 
slope  and  a  correction  applied  to  the  result. 

i.  To  measure  by  keeping  the  chain  level,  the  chain- 
men  should  be  provided  with  a  small  plumb  and  line, 
so  that  the  end  of  the  chain  may  be  held  vertically  over 
the  proper  point.*  If  the  slope  is  not  very  steep,  the 
whole  length  of  the  chain  can  be  laid  off  at  once  by 
raising  the  lower  end  until  the  chain  is  horizontal  and 
transferring  the  end  to  the  ground  with  the  plumb  ;  but 


and  hypotenuse  is  equal  to  the  square  of  the  perpendicular  divided  by  twice 
the  known  side. 

The  degree  of  approximation  involved  in  the  preceding  relation  is  shown 
by  the  following  table  : 

SLOPE.  ERROR. 

5  vertical  to  100  horizontal i  in  1,000,000 


10 

20 

30 
40 

50 
60 
80 

IOO 


100 
IOO 
IOO 
IOO 
IOO 
IOO 
IOO 
IOO 


C.OOO 

1,000 

I  OOO 

6 

1,000 

•£.,;,      12 

I  OOO 

30 

1,000 

.  57 

1,000 

This  approximation  is  frequently  very  convenient,  and  the  student  should  get 
it  well  fixed  in  mind. 

*  The  common  practice  of  dropping  a  chaining-pin  when  one  end  of  the 
chain  is  elevated  only  two  or  three  feet,  although  recommended  by  many  text- 
books, is  objectionable.  The  error  due  to  the  pin's  not  dropping  vertical  is 
probably  greater  than  the  error  to  be  eliminated. 


l8  CHAIN    AND    TAPE.  [CHAP.   I 

when  the  slope  is  so  steep  that  the  two  ends  of  the  chain 
can  not  conveniently  be  brought  to  the  same  horizontal 
line,  then  only  part  of  the  chain  can  be  laid  off  at  a  time, 
in  which  case  some  aliquot  part  should  be  used.  When 
only  part  of  the  chain  is  used,  great  care  must  be  taken 
not  to  confuse  the  count. 

This  method  of  chaining  involves  three  difficulties  : 
(a)  keeping  the  chain  horizontal  ;  (b)  transferring  the 
elevated  end  of  the  chain  vertically  to  the  ground  ;  and 
(c)  making  the  stretch  from  the  pull  equal  to  the  short- 
ening from  sag.  a.  It  is  very  difficult  to  determine  a 
level  line,  particularly  when  one  is  standing  at  one  end 
of  it  and  looking  up  or  down  hill.  In  chaining  up  steep 
slopes  it  is  a  great  advantage  to  have  a  third  man,  who 
shall  stand  at  one  side  of  the  line  and  tell  when  the, 
chain  is  horizontal.  Even  then  one  is  liable  to  be. 
greatly  deceived  as  to  the  position  of  a  horizontal  line. 
Generally  the  apparently  horizontal  line  is  too  nearly 
parallel  with  the  slope,  b.  It  is  nearly  impossible  to 
hold  the  plumb-line  exactly  at  the  end  of  the  chain  and 
keep  the  chain  both  horizontal  and  sufficiently  stretched, 
and  at  the  same  time  hold  all  so  steady  that  the  plumb- 
line  will  hang  still,  c.  The  amount  of  pull  required  to 
overcome  the  shortening  from  sag  can  be  determined 
only  by  trial.  To  do  this,  stretch  the  chain  between 
two  points  at  the  same  elevation,  having  it  supported 
its  entire  length,  and  remove  the  supports,  noting  how 
strong  a  pull  is  required  to  bring  the  ends  of  the  chain 
to  the  marks  again.  This  should  be  done  by  the  chain- 
men  themselves,  to  enable  them  to  judge  how  hard 
to  pull  it  when  it  is  off  the  ground.  If  only  part  of  the 
chain  is  to  be  laid  off  at  once,  this  test  must  be  applied 
to  that  portion  also. 

Obviously  it  is  not  possible  to  perform  all  of  the 
preceding  operations  with  any  considerable  degree  of 
accuracy;  and  it  is  generally  more  expeditious  and  also 


ART.  3]  USING    THE    CHAIN.  19 

more  accurate  to  measure  on  the  slope  and  apply  a  cor- 
rection. 

2.  To  compute  the  difference  between  the  distance  on 
the  slope  and  that  on  the  horizontal  requires  the  deter- 
mination of  the  rate  of  slope.  This  can  be  found  by 
estimation,  by  means  of  a  pocket  level  or  a  clinometer, 
or  by  estimating  the  horizontality  of  a  flag-pole  and 
measuring  down.  The  quantity  to  be  subtracted  from 
the  distance  on  the  slope  is  then  computed  by  equation 
(i),  page  16. 

Notice  that  the  accuracy  of  the  second  method  is  de- 
pendent upon  the  exactness  with  which  the  rate  of  slope 
can  be  determined;  but  as  this  is  only  one  of  the  three 
difficulties  encountered  in  the  first  method,  we  may  con- 
clude that  the  second  is  the  more  accurate.  It  is  also 
the  more  expeditious. 

18.  COMPENSATING  vs.  CUMULATIVE  ERRORS.  Before 
considering  the  several  errors  to  which  chaining  is 
liable,  it  will  be  well  to  notice  that  in  all  measuring 
operations  the  observer  should  carefully  distinguish  be- 
tween two  classes  of  errors;  viz.,  compensating  errors,  or 
those  which  are  as  likely  to  be  plus  as  minus,  and  tend 
to  balance  each  other;  and  cumulative  errors,  or  those 
which  always  have  the  same  sign  and  affect  the  final 
result  in  the  same  way.  This  distinction  is  very  impor- 
tant. The  observer  should  avoid  errors  which  usually 
occur  in  a  single  direction,  but  he  need  not  always  take 
so  great  care  to  avoid  errors  which  are  as  liable  to  be 
negative  as  positive.  An  apparently  inappreciable  but 
cumulative  error  may  in  the  course  of  a  series  of  ob- 
servations amount  to  more  than  a  much  larger  but 
compensating  error.  The  effect  of  compensating  errors 
is  reduced  nearly  to  zero  simply  by  multiplying  the 
number  of  observations;  but  cumulative  errors  should 
be  avoided  entirely,  or  observations  made  by  which 
they  may  be  corrected. 


20  CHAIN    AND    TAPE.  [CHAP.  I 

The  uncertainty  in  the  length  of  a  line  due  to  compen- 
sating or  accidental  errors  varies  as  the  square  root  of 
the  number  of  units  in  the  line,  while  the  effect  of  cu- 
mulative or  constant  errors  varies  directly  as  the  length. 
The  whole  amount  of  the  cumulative  errors  remains  un- 
corrected,  while  only  the  square  root  of  the  compensat- 
ing errors  is  uncompensated.  For  example,  if  the  chain 
is  o.i  of  a  foot  too  long,  a  line  25  chains  long  will  be  re- 
corded 2.5  feet  too  short;  but  if  the  pin  is  sometimes 
set  o.i  of  a  foot  beyond  the  end  of  the  chain  and  some- 
times the  same  amount  behind,  the  final  error  at  the  end 
of  the  line  due  to  this  error  is  probably  only  o.i  ^25  or 
0.5  foot.  If  the  head  chain-man  has  a  fixed  habit  of 
setting  the  pin  beyond  the  end  of  the  chain,  then  this 
becomes  a  cumulative  error,  and  varies  as  the  dis- 
tance. 

This  illustrates  that  in  the  prosecution  of  any  work  it 
is  desirable  that  the  operator  should  be  cognizant  of  the 
nature  and  importance  of  every  source  of  error.  The 
more  thorough  and  complete  his  knowledge  in  this  re- 
spect, the  more  readily  and  accurately  will  he  be  able  to 
decide  what  source  of  error  may  be  wholly  neglected, 
what  may  be  partially  provided  against,  and  what  must 
be  carefully  avoided  or  eliminated.  This  knowledge  is 
conducive  both  to  greater  accuracy  and  to  economy  of 
time  and  effort;  for  the  observer,  knowing  that  what 
might  otherwise  have  been  attended  to  with  consider- 
able care  may  be  neglected,  is  free  to  give  all  his  atten- 
tion to  the  weakest  link  in  the  chain  of  observations. 
It  enables  the  observer  to  correctly  proportion  his  pains 
to  the  degree  of  precision  required.  A  good  observer  is 
one  who  is  able  to  take  just  care  enough  to  attain  the 
desired  accuracy,  without  wasting  time  and  energy  in 
uselessly  perfecting  certain  parts  of  the  work.  He  must 
be  able  to  discover  the  relative  accuracy  required  in 
different  parts  of  a  complete  observation.  All  this  calls 


ART.  3]  USING    THE    CHAIN.  21 

for  a  clear  understanding  of  the  causes  of  error,  and  the 
ability  to  determine  their  effect  upon  the  final  result. 

19.  SOURCES  OF  ERROR.  The  sources  of  error  in  chain- 
ing are  (a)  incorrect  length  of  chain,  (b)  kinking  of  the 
chain  and  bending  of  the  links,  (c)  the  chain's  not  toeing 
in  a  vertical  plane,  (a)  unequal  tension  of  the  chain,  (<?) 
expansion  and  contraction  with  changes  of  temperature, 
(/)  errors  of  lining  the  fore  chain-man,  (g)  not  placing 
the  pin  at  the  end  of  the  chain,  (h)  drawing  the  pin  by 
hanging  the  back  handle  over  it,  (/)  chain  not  being 
level,  and  (/)  such  errors  as  miscounting  tallies  or  chains, 
counting  from  wrong  end  of  chain,  making  a  mistake 
of  10  in  the  number  of  links,  reading  18  for  22,  37  for  43, 
etc.  Errors  of  the  last  class  are  much  too  common, 
but  can  be  obviated  by  care  and  thoughtfulness. 

a.  An  incorrect  length  of  chain  is  a  constant  or  cumu- 
lative error,  and  may  be  plus  or  minus.  When  one  is 
desirous  of  attaining  the  last  degree  of  accuracy,  this 
is  the  most  difficult  error  to  eliminate;  and  even  for 
the  accuracy  required  in  ordinary  surveying,  it  is  an 
important  element.  Recent  experience  of  the  author 
will  illustrate  the  difference  Which  exists  between  so- 
called  standards.  He  had  occasion  to  compare  a  100- 
foot  steel  tape  just  from  the  shop  of  a  reputable  manu- 
facturer, which  was  "  guaranteed  to  be  true  to  U.  S. 
standard,"  with  a  2o-foot  pole  said  to  have  been  pro- 
nounced correct  by  the  "  U.  S.  A.  engineers,"  and  also 
with  a  standard  derived  with  inappreciable  error  from 
two  2-foot  steel  rules  made  by  the  best  tool-makers  in 
the  United  States.  The  errors  of  the  intercomparisons 
were  inappreciable.  The  tape  was  0.95  of  an  inch  short 
by  the  first,  while  by  the  second  it  was  0.45  of  an  inch 
short.  A  subsequent  comparison  at  the  office  of  the 
Mississippi  River  Commission  showed  the  tape  to  be 
0.256  ±  0.07  of  an  inch  short.  The  above  differences 
were  practically  independent  of  temperature  correction. 


22  CHAIN    AND    TAPE.  [CHAP.  I 

After  the  chain  has  once  been  adjusted  to  the  stand- 
ard, it  should  be  tested  frequently.  The  common  chain 
has  600  wearing  surfaces,  and  if  each  wears  only  o.oi  of 
an  inch,  the  length  is  increased  6  inches.  A  tempered- 
steel  chain  with  brazed  links  lengthened  half  an  inch 
in  chaining  70  miles.  It  is  not  uncommon  to  find  chains 
differing  i,  2,  or  even  3  inches. 

A  steel  tape  is  liable  to  have  its  length  changed  each 
time  it  is  repaired,  and  is  also  liable  to  be  permanently 
lengthened  by  excessive  pull  in  using,  and  by  hammering 
it  to  straighten  out  short  bends.  Therefore  the  steel 
tape,  as  well  as  the  chain,  should  be  tested  occasion- 
ally. 

For  the  method  of  applying  a  correction  to  reduce 
the  chain  to  the  true  standard,  see  §  15. 

b.  Kinking  of  the  chain  and  bending  of  the  links  are 
sources  of  plus  cumulative  errors;  i.e.,  they  tend  to  make 
the  recorded  distance  too  great.  Mud  and  ice  in  the 
joints  produce  an  error  in  the  same  direction,  while  the 
opening  of  the  rings  produces  an  error  in  the  opposite 
direction.  The  net  result  may  be  either  plus  or  minus. 
This  source  of  error  is  easily  avoided  by  substituting  a 
steel  tape  or  wire  for  the  chain. 

C.  If  the  chain  is  not  stretched  straight,  the  resulting 
error  is  cumulative,  and  tends  to  make  the  recorded 
distance  too  great.  If  the  center  of  a  roo-foot  chain 
is  i  foot  out  of  line,  the  error  is  0.02  ft.,*  and  varies  as 
the  square  of  the  error  of  alignment;  that  is,  if  the 
middle  point  is  half  a  foot  out  of  line,  the  error  in  dis- 
tance is  only  0.005  ft.f  The  time  required  to  get  the 
chain  straight  between  pins  can  be  greatly  lessened  by 


*  See  foot-note  page  16. 

t  Let  the  student  show  that  if  a  point  not  the  middle  of  the  chain  is  out 
given  amount,  the  error  is  a  little  greater  than  if  the  center  were  out  a  li 
amount, 


ART.  3]  USING    THE   CHAIN. 


the  fore  chain-man's  being  careful  to  walk  on  the  line 
being  measured. 

d.  The  effect  of  a  difference  in   the  pull  on  the  tape 
or  chain  is  compensating.     For  the  tape  this  error  varies 
directly  as  the  difference  of  pull,  directly  as  its  length, 
and  inversely  as  its  cross-section.     To  find  the  elonga- 
tion of  a  steel  tape,  represent  its  length  in  inches  by  Z, 
and  its  cross-section   in   square  inches   by  S;  then  the 

elongation  for  a  pull  of  one  pound  is  ---  ~  .    For 

30,000,0000 

a  loo-foot  tape  0.2  inch  wide  by  0.02  inch  thick,  this 
gives  an  elongation  of  o.oi  of  an  inch  per  pound  of 
pull.  Tapes  are  sometimes  made  with  a  spring  balance 
attached  at  one  end,  whereby  this  source  of  error  can 
be  practically  eliminated. 

The  elongation  of  a  chain  will  depend  mainly  upon 
the  form  and  size  of  the  rings  which  connect  the  sev- 
eral links  together,  and  can  be  determined  only  by 
trial  with  a  spring  balance. 

e.  The   error  due   to   expansion  and   contraction   is 
generally  cumulative,  and  may  be  either  plus  or  minus. 
Tapes  are  usually  made  standard   at  60°  Fahr.     For  a 
loo-foot  tape   a  change  of  30°  F.  makes  a  difference  of 
one  fourth  of  an  inch.*     By  remembering  this  relation 
it  is  easy  to  determine  the  tape  correction  at  any  tem- 
perature.    For  example,  if  the  tape  is  true  at  60°,  then 
at  90°  the  tape  will  be  0.02  ft.  (one  quarter  of  an  inch) 
too  long;  i.e.,  at  90°  the  tape  is   1.0002  times  the  stand- 
ard length.     At   30°    the   length    of    the    tape    will   be 
0.9998   (=  i  —  .0002)    times    the    standard.      In    ordi- 
nary work  it  is  only  the  extremes  of  temperature  that 
will  cause  the  expansion  and  contraction  of  the  tape  to 


*  The  co-efficient  of  expansion  varies  with  the  kind  and  quality  of  the  steel, 
but  closely  approximates  0.0000065  per  unit  of  length  per  i°  F, 


24  CHAIN    AND   TAPE.  [CHAP.  I 

be  appreciable;  but  in  accurate  work  this  is  one  of 
the  chief  sources  of  error,  and  one  very  difficult  to 
eliminate.  Tapes  intended  for  city  surveying  and 
other  accurate  work  sometimes  have  a  thermometer 
attached,  by  means  of  which  to  determine  the  cor- 
rections for  temperature;  but  even  this  is  not  a  perfect 
remedy,  since  when  the  sun  is  shining  the  temperature 
of  the  tape  is  often  quite  a  good  deal  higher  than 
that  of  the  atmosphere.  There  are  several  other,  but 
minor,  difficulties  to  be  overcome  in  eliminating  this 
error. 

f.  The  error  due  to  not  setting  the  fore  pin  exactly 
in  line  is  generally  overestimated  in  proportion  to  the 
other  errors.  An  undue  amount  of  time  is  therefore 
given  to  lining  in  the  fore  chain-man.  With  a  loo-foot 
chain,  if  the  pin  is  i  foot  out  of  line,  the  error  is  0.005 
ft.  (one  sixteenth  of  an  inch).  The  error  varies 
directly  as  the  square  of  the  error  of  alignment  and 
inversely  as  the  length  of  the  chain.  Consequently  the 
longer  the  tape  the  less  the  error  from  this  cause,  and 
the  greater  the  speed.  The  error  is  cumulative  and 
plus. 

Vertical  inequalities  in  the  ground  produce  errors 
similar  to  those  in  aligning  the  fore  chain-man,  and 
unfortunately  in  many  cases  this  element  limits  the 
degree  of  accuracy  attainable.  Supporting  the  tape  by 
hand  and  plumbing  down  is  of  very  little  advantage. 
This  source  of  error  can  be  eliminated  by  suspending 
the  tape  between  fixed  supports.  The  attempt  has  fre- 
quently been  made  to  introduce  such  devices  for  city 
surveying,  but  they  have  not  met  with  general  favor. 
They  are  complicated,  cumbersome,  expensive  to  oper- 
ate, and  contribute  little,  if  any,  to  accuracy.  The 
chief  source  of  error  in  all  such  devices  that  the 
writer  has  seen  is  in  plumbing  down  from  the  end 
of  the  tape.  The  effect  of  the  wind  upon  the  sag  is 


ART.  3]  USING    THE   CHAIN.  25 

another  important  source  of  error.  Such  devices  are 
appropriate  for  the  measurement  of  geodetic  base  lines. 
For  an  illustrated  account  of  a  simple,  but  accurate, 
method  of  using  a  suspended  tape,  see  Annual  Report 
of  the  Missouri  Commission  for  1886 — Executive  Doc- 
ument No.  28,  49th  Congress,  2d  Session, — pp.  31-35. 

g.  Many  chain-men  hold  the  pin,  while  setting  it,  in 
such  a  manner  that  the  point  enters  the  ground  consid- 
erably in  front  of  the  end  of  the  chain;  then  when  the 
rear  end  is  brought  forward  it  is  laid  on  the  ground, 
thus  introducing  considerable  error.  This  error  may 
occur  with  a  tape,  but  is  usually  in  the  opposite  direction. 
The  remedy  is  obvious. 

The  very  general  practice  of  placing  the  forward 
handle  of  the  chain  against  one  side  of  the  pin  and  the 
back  handle  against  the  other  side,  can  not  be  consid- 
ered precise.  The  same  side  of  the  pin  should  be  used 
both  times;  then  the  length  of  the  chain  is  from  outside 
of  one  handle  to  the  inside  of  the  other.  A  similar  crit^- 
icism  is  applicable  to  the  way  many  steel  tapes  are  used 
(see  the  last  paragraph  of  §  3). 

h.  The  error  due  to  pulling  the  pin  over  after  it  is  set 
is  usually  cumulative,  and  may  be  either  plus  or  minus. 
It  may  be  entirely  eliminated  by  care.  The  back  handle 
should  not  be  dropped  over  the  pin  until  the  chain  is  in 
position  preparatory  to  laying  off  the  distance.  With  a 
properly  made  tape  (see  the  last  paragraph  of  §  3)  this 
particular  source  of  error  can  be  entirely  eliminated; 
but  even  then  the  rear  chain-man  must  be  careful  that  he 
does  not  push  the  back  pin  over  when  holding  the  tape 
against  it. 

i.  The  amount  of  error  due  to  the  chain's  not  being 
horizontal  and  the  method  of  reducing  the  resulting 
error  have  already  been  discussed  in  §  17. 

20.  Let  us  see  if  we  can  determine  the  final  error 
due  to  an  assumed  value  for  each  of  the  above  sources 


26  CHAIN   AND   TAPE.  [CHAP.  I 

of  error.  Notice  that,  under  ordinary  conditions,  all  the 
errors  are  cumulative  except  d  and  g.  If  the  line  to  be 
measured  is  n  chains  long, 

the  final  error  = 

n(±  a±b  +  c±e+f±h±t)+Vn(±d±g).      (i) 


According  to  the  theory  of  probability,  equation  (i) 
becomes, 

the  probable  final  error  = 


W  (a*  +  &  +  S  +  S+f  +  #  +  ,*)  +  n  (<?  +/).     (2) 

To  illustrate  the  method  of  using  this  formula,  assume 
that  a  line  1,000  feet  long  is  to  be  measured  with  a  steel 
tape.  Assume  that  the  several  partial  errors  are  as  fol- 
lows: 0=0.01;  ^  =  0.0;  *=o.oi;  */=o.o2;  *=o.oi: 
/=  0.005;  £-=0.01;  ^  =  0.0;  /=o.oi.  Substituting 
these  values  in  equation  (2),  and  solving, 

the  probable  final  error  —  0.22  ft     ...      (3) 

The  student  should  carefully  consider  the  degree  of 
care  necessary  to  reduce  the  several  errors  to  the  values 
assumed  above.  Are  the  above  values  correct  relatively? 
i.e.,  do  they  represent  equal  care  in  the  several  opera- 
tions ?  Are  the  errors  correctly  classified  as  cumulative 
and  compensating  ? 

Notice  that  if  several  measures  of  a  line  made  with 
the  same  tape  are  to  be  compared,  the  error  a  should  not 
be  included  in  the  above  equations.  Notice  also,  that 
if  the  reduction  for  grade  is  computed  by  the  second 
method  of  §  17  (page  19),  the  error  /  should  not  be  in- 
cluded. Notice  again,  that  if  the  same  men  measure  the 
line  with  the  same  appliances  under  the  same  condi- 
tions, the  probable  error  deduced  from  equation  (2) 


ART.  3]  USING    THE    CHAIN.  27 

should  be  greater  than  that  deduced  from  the  measure- 
ments of  the  distance,  for  the  latter  involves  only  the 
variation  in  the  errors,  while  the  former  depends  upon 
the  errors  themselves.  See  Problem  No.  i,  Appendix 
IV. 

21.  LIMITS  OF  PRECISION.     It  is  very  desirable  that  each 
engineer  should  know  the  uncertainty  of  his  ordinary 
work.     If  this  could  be  definitely  stated  for  each  method 
of  measuring  and  for  the  different  kinds  of  ground,  it 
would  be  very  instructive;  but  an   engineer  can  learn 
only  by  experience  the  amount  of  care  and  time  required 
to  attain  any  particular  degree  of  accuracy.     The  labor 
required  increases  more  rapidly  than  the  degree  of  pre- 
cision attained,  and   the   more  accurate  the  work  the 
greater  the  difference. 

A  few  words  are  needed  as  to  the  difference  between 
real  and  apparent  errors.  If  a  line  is  twice  measured 
with  the  same  chain  by  the  same  men  under  the  same 
conditions,  the  difference  between  the  two  measure- 
ments represents  the  difference  between  the  accidental 
errors  each  time,  and  does  not  show  the  real  error  in  the 
observed  length.  Any  constant  or  cumulative  error,  as 
an  incorrect  length  of  the  chain,  would  be  the  same  in 
each.  This  distinction  is  very  important  in  discussing 
the  errors  of  any  series  of  observations. 

22.  The    following   data    are    given     to    assist   the 
student  in  forming  his  standard  of  good  work.     Unfor- 
tunately, there  is  very  little  on  this  subject  to  be  found 
in  engineering  literature. 

Burt*  found  in  the  early  surveys  of  the  U.  S.  public 
lands,  that  for  common  timber-land  "with  two  sets  of 
chainmen  instructed  alike  in  the  proper  manner  of 
keeping  their  chain  level  and  straight  on  the  line,  and 
of  setting  the  tally  pins  plumb,  as  well  as  holding  the 

*  Key  to  the  Solar  Compass,  p.  35. 


28  CHAIN    AND    TAPE.  [CHAP.   I 

ends  of  the  chain  to  them,  the  average  difference  was 
i  in  500,  and  sometimes  i  in  220,  and  under  the  most 
favorable  conditions  it  was  i  in  1,600.*' 

At  present  the  standard  for  good  chaining  in  survey- 
ing the  public  lands  of  Canada  is  a  difference  of  i  in 
5,300  for  two  sets  of  men  chaining  the  same  line  under 
the  same  conditions.*  "  Experience  with  very  careful 
men  shows  that  chaining  on  level  prairie  is  about  two 
links  to  a  mile  (i  in  4,000)  longer  than  over  hilly  and 
broken  prairie,  or  over  windfall  and  brush  in  the 
woods."  * 

The  author's  students,  in  ordinary  class  work  in  land 
surveying,  measure  lines  from  200  to  1,000  feet  long, 
with  a  66-foot  chain,  and  repeat  with  a  difference  of 
i  in  15,000  to  i  in  20,000  for  the  same  men;  for  dif- 
ferent men  on  different  days  with  different  chains 
(compared,  however,  with  the  same  standard),  the  max- 
imum difference  is  i  in  800  to  i  in  1,000,  the  average 
difference  being  i  in  3,000  to  i  in  4,000.  The  ground 
is  favorable,  but  the  work  is  done  with  the  ordinary 
expedition  of  actual  practice.f  The  same  students, 
after  a  term's  practice,  with  a  loo-foot  steel  tape  meas- 
ure lines  500  to  1,000  feet  long  and  repeat  with  a 
maximum  probable  error  for  the  surface  distance  of  i 
in  40,000,  and  an  average  of  i  in  103,000  for  the  same 
men;  and  for  different  men  on  different  days  with  dif- 
ferent tapes  (compared  with  the  same  standard),  the 
maximum  error  is  i  in  5,000,  and  the  average  error  i 

*  Report  of  the  Proceedings  of  the  Association  of  Dominion  Land  Survey- 
ors for  1890,  p.  58. 

t  All  values  of  the  degree  of  precision  credited  in  this  volume  to  the  au- 
thor's students  are  results  attained  in  the  ordinary  class  work  of  the  first  or 
second  term's  field  practice.  Usually  the  value  given  is  the  mean  for  the 
whole  class.  The  results  are  obtained  from  a  specially  prepared  area  the  di- 
mensions of  which  are  accurately  known.  The  lines  vary  from  30  to  800  feet, 
and  the  areas  from  0.5  to  7  acres,  the  longer  lines  and  the  larger  areas  occur- 
ring more  frequently  in  the  problems. 


ART.  3]  USING    THE    CHAIN.  29 

in  14,300.  For  the  purpose  of  this  record,  eleven  pairs 
of  chainmen  on  different  days  measured  a  line  1,000 
feet  long  with  a  probable  error  for  a  single  measure- 
ment of  the  surface  distance  of  i  in  7,800;  omitting  one 
result,  the  probable  error  is  i  in  10,800.  The  difference 
of  elevation  was  17  feet,  and  the  reduction  for  grade 
was  0.288,  as  determined  from  an  accurate  profile  of 
the  line;  and  the  probable  error  of  a  single  estimate  of 
the  correction  for  level  (by  equation  (i),  page  16,  the 
student  having  had  no  experience  in  leveling)  was  i  in 
1 1, 600.  The  probable  error  of  a  single  determination 
of  the  horizontal  distance  was  i  in  6,500,  and  omitting 
one  result  the  error  was  i  in  7,400. 

Seventy  miles  of  the  finished  track  of  the  Illinois 
Central  R.  R.  was  measured  in  1876  with  a  loo-foot  steel 
chain,  and  re-measured  in  1885  with  a  loo-foot  steel 
tape,  with  a  difference,  after  correcting  for  the  differ- 
ence of  standard  and  applying  a  correction  for  wear  of 
the  chain,  on  the  average  for  the  different  miles  of  i  in 
2,360. 

On  the  Atchison,  Topeka  and  Santa  Fe  R.  R., 
through  Western  Kansas,  the  difference  between  the 
preliminary  survey  and  the  government  land  survey 
averaged  about  i  in  1,000;  and  the  difference  between 
the  preliminary  survey  and  the  location  was  about  i  in 
2,500. 

The  U.  S.  army  engineers  in  re-measuring  the  Union 
and  Central  Pacific  railroads  with  steel  chains  found 
the  error  of  their  own  work  to  average  i  in  23,600,  the 
maximum  being  i  in  10,000,  as  determined  by  retracing 
distances  varying  from  2,000  to  26,000.* 

On  the  U.  S.  Lake  Surveyf  base  lines  for  topograph- 
ical surveys  were  measured  with  a  chain  20  meters 

*  House  Executive  Document,  No.  37,  2d  Session,  44th  Congress,  pp.  14, 
21,  37- 
t  Report  of  Chief  of  Engineers  U.  S.  A.,  1876,  part  III,  p.  9. 


30  CHAIN    AND    TAPE.  [CHAP.  1 

long,  which  differed  from  the  ordinary  chains  only  in 
being  made  of  heavier  wire  and  having  links  20  inches 
long.  In  15  lines,  the  maximum  error  was  i  in  5,700, 
the  mean  being  i  in  17,500,  and  the  minimum  i  in 
45,100. 

In  connection  with  the  U.  S.  Lake  Survey  work,  a 
line  12  miles  long  was  measured  on  the  railroad  track 
with  a  wire  150  feet  long,  at  the  rate  of  about  10  miles 
per  day,  with  a  difference  between  two  measurements 
of  i  in  32,000. 

The  U.  S.  Coast  and  Geodetic  Survey  Report  for 
1882,  page  191,  contains  an  account  of  the  preliminary 
measurement  of  two  base  lines  with  an  iron  wire  J-  of 
an  inch  in  diameter,  200  feet  long,  in  which  the  differ- 
ence, as  compared  with  the  measurement  of  the  geo- 
detic base  apparatus,  was  i  in  30,000  and  i  in  28,000. 


CHAPTER  II. 

TRIPOD,    LEVELING  SCREWS,    AND  PLUMB-BOB. 

ART.  1.     THE  TRIPOD. 

23.  CONSTRUCTION.  The  manner  of  connecting  the 
leg  with  the  head  needs  attention.  The  leg  should  not 
be  placed  between  two  lugs  or  ears  fastened  to  the 
plate,  for  in  case  the  leg  wears  or  shrinks  there  is  no 
adequate  means  of  making  it  fit.  Drawing  the  ears 
together  by  means  of  the  screws  through  the  top  of  the 
leg  bends  the  plate,  and  even  then  only  partially  reme- 
dies the  evil.  An  excellent  form  is  that  in  which  the 
leg  is  made  of  two  pieces  that  bear  upon  opposite  sides 
of  a  lug  cast  upon  the  plate.  Sometimes  the  leg  is  in 
one  piece  with  a  slot  at  the  top,  which  is  also  very 
good.  Another  good  form  is  that  in  which  the  leg  is 
not  open  at  the  top,  but  bears  upon  only  one  side  of  the 
lug.  In  the  three  forms  last  mentioned  any  looseness 
is  taken  up  by  a  thumb-nut.  Sometimes  the  leg  is  in- 
serted in  a  metal  cap,  which  is  then  fastened  to  the  tri- 
pod head.  This  is  better  than  putting  the  leg  directly 
between  two  metal  ears,  but  is  inferior  to  all  the  other 
forms.  No  instrument  can  stand  firmly  if  there  is  any 
looseness  in  fitting  of  legs  or  shoes. 

The  best  instrument  makers  construct  the  tripod  legs 
of  two  pieces  braced  together,  or  of  a  solid  piece  cut 
out  in  such  a  way  as  to  lighten  it  without  materially 
affecting  its  strength. 

Some  instrument  manufacturers  make  a  tripod  with 

31 


32     TRIPOD,  LEVELING    SCREWS,  AND    PLUMB-BOB.      [CHAP.  II 

legs  that  may  be  made  longer  or  shorter  as  circum- 
stances require.  In  surveying  in  a  mine  or  tunnel  this 
is  a  great  convenience,  if  not  a  necessity. 

24.  SETTING  THE  TRIPOD.  In  setting  the  tripod  notice 
that  to  alter  the  position  of  the  plumb-line  the  legs 
must  be  swung  on  their  pivots,  but  not  sidewise;  and 
also,  that  to  get  the  plate  level  the  legs  must  be  swung 
sidewise,  but  not  on  their  pivots.  These  two  points, 
though  seemingly  trivial  in  themselves,  are  worth 
remembering  as  a  means  of  saving  time  and  labor,  and 
also  of  preventing  unnecessary  wear  and  strain  on  the 
leveling  screws,  which  last,  owing  to  faulty  construction 
of  the  leveling  appliances,  is  frequently  very  important. 
Attention  to  the  following  principle  also  will  save 
much  time  and  hard  labor  in  setting  the  tripod.  Let 
the  lines  radiating  from  cy  Fig.  4,  represent  lines  join- 
ing the  feet  of  the  tripod  legs  with 
the  point  over  which  the  plumb-bob 
is  to  be  placed,  and  b  the  position 
of  the  plumb-bob.  It  is  desired  to 
move  the  plumb-bob  from  b  to  c. 
Press  the  leg,J?  into  the  ground  until 
the  bob  swings  to  a  in  the  line  3^  pro- 
longed, and  then  force  leg  3  into  the 
ground  until  the  bob  swings  to  c.  In  general,  press  one 
of  the  legs  into  the  ground  until  the  plumb-bob  swings 
to  the  side  of  the  point  opposite  one  of  the  other  legs; 
then  press  that  leg  in  until  the  plumb-bob  arrives  at 
the  center. 

The  tripod  should  be  set  firmly,  but  a  great  deal  of 
time  and  effort  is  frequently  wasted  in  forcing  the  legs 
into  the  ground  needlessly.  Setting  the  tripod  is  an 
operation  that  must  be  repeated  many  times>  and  the 
beginner  should  learn  to  do  it  quickly  and  easily. 


ART.   2]  LEVELING    SCREWS.  33 


ART.  2.     LEVELING  SCREWS. 

25.  Most  instruments  are  provided  with  four  leveling 
or  foot  screws,  while  others  have  only  three.     The  in- 
strument can  be  leveled  more  quickly  with  three  screws 
than  with   four,  since  by  using  both  hands   the  instru- 
ment  can   be  leveled    in   both   directions  at   the   same 
time.      Three    screws    are    also     more    sensitive,   and 
are    less   liable   to    strain  and  damage   the  instrument 
than    four — an  important  consideration  with  many  in- 
struments, owing    to    faulty  construction  of  the    level- 
ing  appliances.     Although   the  best   American  instru- 
ments, and    many  if   riot    most    European    ones,    have 
only  three  foot  screws,  nearly  all  American  engineering 
instruments  have  four.     Doubtless  this  is  partly  due  to 
custom,  and    partly    to    the   fact    that   for   mechanical 
reasons  it  is  a  little  easier  to  arrange  the  shifting  plate 
of  the   transit  (§  in)  with  four  than   with   three  foot 
screws. 

Whatever  the  number,  there  should  be  no  looseness 
between  the  screw  and  the  nut.  This  looseness  is  par- 
ticularly objectionable  in  a  leveling  instrument  or  in  a 
transit  used  to  measure  vertical  angles.  To  insure 
steadiness  the  leveling  screws  should  work  in  the  arms 
of  a  solid  star-shaped  casting,  instead  of  in  a  thin 
round  plate  into  which  the  nuts  are  simply  stuck — as  is 
usual.  The  end  of  the  arm  should  be  split  and  pro- 
vided with  a  clamp  screw  by  which  to  adjust  for  wear. 

26.  A  very  serious  defect  in   many  instruments  with 
four   leveling   screws   is    that   the   lower   ends    of   the 
screws   are   above    the   center   of   the    ball-and-socket 
joint   which    fastens    the    upper    part   of   the    instru- 
ment to  the  tripod  (see  Fig.  24,  page  95).     Then   when 
one  pair  of  screws  is  being  used  to  level  the  plate,  the 
upper  part  of  the  instrument  must  revolve  about  a  line 


34     TRIPOD,  LEVELING    SCREWS,  AND    PLUMB-BOB.      [CHAP.  II 

parallel  to,  but  below,  a  line  joining  the  feet  of  the  pair 
of  screws  not  in  use;  therefore  the  instrument  can  not 
revolve  without  causing  the  screws  not  in  use  to  bind, 
and  it  can  turn  only  by  causing  the  feet  of  these 
screws  to  slip  on  the  lower  plate.  Besides  the  annoy- 
ance in  leveling  the  instrument,  this  binding  tends  to 
bend  the  screws  and  warp  the  plates;  and  the  slipping 
defaces  the  instrument  by  cutting  spherical  holes  in 
the  lower  plate.  Placing  the  feet  of  the  screws  in 
small  cups  prevents  the  holes  in  the  upper  face  of  the 
lower  plate,  but  increases  the  objections  in  the  other 
and  more  important  respects.  Providing  the  foot  of 
the  leveling  screw  with  a  ball-and-socket  joint  is  no 
improvement  over  the  simple  cup,  except  to  prevent  the 
cup  (socket)  from  getting  lost. 

The  above  defect  may  be  remedied  by  bringing  the 
center  of  the  ball-and-socket  joint  into  the  plane  of  the 
feet  of  the  leveling  screws.  Since  with  three  leveling 
screws  the  instrument  can  not  be  attached  to  the  tripod 
by  a  ball-and-socket  joint,  this  defect  can  not  exist  in 
that  form  of  construction.  With  three  leveling  screws 
the  instrument  is  fastened  to  the  tripod  by  a  spiral  spring. 

Since  nearly  all  instruments  have  the  above  defect, 
it  is  very  necessary  that  they  should  be  leveled  approxi- 
mately by  manipulating  the  tripod  legs  as  already  de- 
scribed (§  24). 

27.  Some  instruments  are  provided  with  an  arrange- 
ment for  approximately  leveling  the  instrument  very 
quickly  without  manipulating  the  foot  screws,  of  which 
there  are  several  forms  on  the  market.  A  quick-leveling 
device  is  specially  suitable  for  leveling  instruments,  but 
by  using  a  little  care  in  setting  the  tripod  it  can  be  dis- 
pensed with  easily.  All  such  additions  are  an  advantage 
in  the  greater  convenience  they  afford,  but  a  disadvan- 
tage in  the  increased  weight,  complexity,  and  cost  in- 
volved, 


ART.  3] 


PLUMB-BOB. 


35 


ART.  3.     PLUMB-BOB. 

28.  The  string  employed  should  be  small  but  strong. 
The  woven  ones  are  best,  since  they  do  not  twist  or 
untwist.  The  string  should  be  connected  to  the  in- 
strument by  a  hook,  so  as  to  be  easily  attached  and  de- 
tached ;  but  it  should  be  suspended  in  such  a  way  that 
the  vertical  of  the  plumb-line  will  always  pass  through 
the  center  of  the  instrument,  however  much  the  lower 
plate  may  be  out  of  the  horizontal. 


K 


a  be 

FIG.  5. — METHODS  OF  ADJUSTING  LENGTH  OF  PLUMB-LINE. 

To  allow  for  an  adjustment  in  the  length  of  the 
plumb-line,  the  free  end  of  the  line  may  be  passed 
through  the  attachment  on  the  instrument  and  then 
knotted  around  the  suspended  portion,  the  adjustment 
being  obtained  by  sliding  the  knot  up  and  down  the 
string.  A  better  method,  particularly  with  a  small  string, 
is  to  pass  it  through  a  small  strip  of  wood,  leather,  or 
metal,  as  shown  at  a,  Fig.  5;  or,  the  end  may  be  fastened 


36     TRIPOD,  LEVELING    SCREWS,  AND    PLUMB-BOB.      [CHAP.  II 

to  a  wire  as  shown  at  b,  Fig.  5.  The  latter  is  the  better, 
since  less  surface  is  exposed  to  the  action  of  the  wind. 
The  adjustable  plumb-bob  shown  at  c,  Fig.  5,  is  still 
better  but  more  expensive.  This  plummet  has  a  con- 
cealed reel  ^,  around  which  the  string  is  wound  by  turn- 
ing the  milled  head  K.  The  friction  of  the  cord  through 
the  guide  /  holds  the  bob  at  any  desired  point  on  the 
line.  Another  form  of  the  last  consists  of  a  short  spool 
on  top  of  the  plumb-bob,  around  which  the  plumb-line 
is  wound.  The  spool  is  turned  by  hand,  and  held  in 
any  position  by  friction. 

The  body  of  the  plumb-bob  should  be  a  long  slender 
cone,  rather  than  a  short  thick  one,  so  that  the  point 
may  be  seen  without  stooping.  The  tip  of  the  bob 
should  be  of  steel,  for  durability.  In  case  the  pointed 
plumb-bob  is  lost,  and  only  a  rough  piece  of  some 
heavy  substance  can  be  had,  the  instrument  may  still  be 
plumbed  down  accurately,  by  holding  a  second  plumb- 
line  before  the  eye  in  such  a  position  that  the  eye 
shall  be  in  the  same  plane  with  the  two  lines ;  then, 
without  moving  the  eye,  have  an  assistant  mark  a  line 
under  the  instrument  in  this  plane.  Repeat  the  opera- 
tion at  90°  from  the  first  position.  The  intersection  of 
these  two  lines  is  the  desired  point. 

Owing  to  defective  casting,  the  plumb-bob  may  not 
hang  vertically,  although  of  true  form  and  apparently 
solid.  In  very  accurate  work  this  might  cause  error. 
A  rough  method  of  testing  this  is  to  hold  the  string  in 
the  hand  and  twist  it  a  little,  and  while  the  string  is 
untwisting,  lower  the  point  into  a  basin  of  water.  If 
the  weight  is  not  truly  distributed  and  consequently 
the  plummet  not  true,  the  eccentric  motion  of  the  point 
will  scatter  the  water. 

If  it  is  desired  to  plumb  down  from  a  high  point,  in- 
stead of  using  a  plumb-line,  which  is  liable  to  be  dis- 
turbed by  the  wind,  it  is  better  to  use  a  transit  accord- 
ing to  a  method  to  K~  ^nlained  farther  on  (§  125). 


CHAPTER  III. 
MAGNETIC  COMPASS. 

ART.  1.     CONSTRUCTION. 

29.  THE  ordinary  form  of  the  magnetic  compass  is 
shown  in  Fig.  6. 

The  value  of  the  magnetic  compass  as  an  engineer- 
ing instrument  depends  chiefly  upon  (i)  the  delicacy  of 
the  needle,  and  (2)  the  constancy  with  which  it  assumes 
the  direction  of  the  magnetic  meridian.  The  first  de- 
pends upon  the  ease  with  which  the  needle  turns  on  the 
pivot,  and  the  second  upon  the  intensity  of  the  directive 
force.  To  satisfy  the  first  condition,  there  is  attached 
to  the  center  of  the  needle  an  agate  cup  to  receive  the 
point  on  which  it  turns;  and  the  pivot  is  made  of  the 
hardest  steel  ground  to  a  smooth,  round,  sharp  point. 
The  second  condition  is  satisfied  by  making  the  needle 
of  shear  steel  and  strongly  magnetizing  it. 

The  needle  should  be  of  such  a  length  that  it  shall 
just  clear  the  graduation,  otherwise  there  will  be  diffi- 
culty in  reading.  Beyond  a  certain  length  (about  5  or  6 
inches)  no  additional  power  is  gained  by  increasing  the 
length  of  the  needle,  owing  to  the  formation  of  second- 
ary poles  in  long  needles.  It  should  not  come  to  rest 
too  quickly;  for  if  it  does,  it  indicates  either  that  the 
needle  is  weakly  magnetized  or  that  the  friction  on  the 
pivot  is  great.  The  needle  should  be  so  sensitive  that 
when  it  is  drawn  to  one  side  by  the  attraction  of  a  piece 

37 


MAGNETIC    COMPASS. 


[CHAP,  in 


of  iron,  it  will  settle  to  the  same  reading  several  times 
in  succession. 


A  compass  is  usually  provided  with  a  vernier  (Chap- 
ter V)  for  setting  off  the  magnetic  declination.     This  is 


ART.   l]  CONSTRUCTION.  39 

indispensable  in  re-surveying  land  originally  laid  out 
with  reference  to  the  magnetic  meridian;  but  it  is  not 
requisite  where  the  land  is  laid  out  according  to  the 
U.  S.  public  land  system. 

Ordinarily  compasses  have  upon  the  edges  of  the 
sights  graduations  such  that  angles  of  elevation  and 
depression  can  be  measured.  This  arrangement  is  par- 
ticularly valuable  for  measuring  the  angle  of  slopes  in 
applying  the  correction  to  reduce  slope  measurements 
to  the  horizontal  distance. 

30.  Extras.     Compasses  are  sometimes  provided  with 
a  graduation  for  reading  angles  independently  of  the 
needle.     This  is  an  important  addition;  but  the  degree 
of   accuracy  attainable  is    limited   by  the  precision  of 
sighting,  which  with  the  open  sights  is  not  very  great. 

Magnetic  compasses  are  sometimes  provided  with  a 
telescope  instead  of  sights.  This  reduces  the  error  of 
sighting,  and  increases  the  length  of  sight ;  but  on  the 
whole  the  advantage  is  not  very  great.  The  chief  mer- 
its of  the  compass  are  cheapness,  portability,  and  rapid- 
ity and  facility  with  which  it  may  be  used.  It  never 
can  be  an  instrument  of  great  accuracy;  and  hence  the 
telescope  adds  cost,  weight,  and  complication,  without  a 
compensating  accuracy.  If  greater  accuracy  is  desired 
than  that  attainable  with  the  ordinary  magnetic  com- 
pass, use  the  transit. 

Some  British  instruments  have  the  graduation  on  a 
ring  attached  to  the  needle  and  moving  with  it,  the 
angles  being  read  by  a  point  projecting  from  the  com- 
pass box.  There  is  no  advantage,  but  great  disadvan- 
tage, in  this  arrangement,  owing  to  increased  weight  on 
the  pivot. 

31.  PRISMATIC  COMPASS.     The    peculiarity  of   this   in- 
strument is  that  a  triangular  glass  prism  is  substituted 
for  one  of  the  sights.     In  looking  through  the  prism  the 
distant  point  and   the  graduation    of   the  compass  are 


MAGNETIC    COMPASS.  [CHAP.  Ill 


visible  at  the  same  time.  The  graduation  is  upon  a 
ring  attached  to  the  needle  and  moving  with  it,  and  the 
bearing  is  read  by  a  point  projecting  from  the  compass- 
box,  under  the  prism.  A  mirror  is  attached  to  the 
sight  in  such  a  way  as  to  reflect  into  the  prismatic 
eye-piece  points  above  or  below  the  horizontal  plane 
of  the  instrument.  The  prism  is  sometimes  worked 


FIG.  7.— PRISMATIC  COMPASS. 

convex  on  the  two  faces  at  right  angles  to  each  other, 
so  as  to  magnify  the  graduation,  and  is  moved  up  and 
down  to  focus  it  (see  Fig.  7).  The  instrument  is  some- 
times provided  with  colored  glass  shades  for  observing 
the  sun. 

Prismatic  compasses  vary  from  2^  to  6  inches  in 
diameter.  They  are  generally  held  in  the  hand,  al- 
though sometimes  mounted  upon  a  Jacob's  staff.  The 
prismatic  compass  is  used  in  preliminary  reconnois- 
sances,  in  clearing  out  lines,  in  "  filling  in  "  in  topo- 
graphical surveying,  etc. 


ART.  2]  TESTS   OP    THE   COMPASS.  41 


ART.  2.     TESTS  OF  THE  COMPASS. 

32.  THE  NEEDLE.      The  needle  should  be  strongly  magnet- 
ized.    If    the  needle    is    not    strongly    magnetized,    the 
directive   force  will  not   overcome  the   friction  on  the 
pivot,  and  hence  the  needle  will  not  take  its  proper  di- 
rection.    Needles  may  be  re-magnetized  with  bar  mag- 
nets, as  described  in  the  text-books  on  physics,  but  such 
methods  are  tedious,  and  give  unsatisfactory  results.     A 
needle  may  be  quickly  and  thoroughly  saturated  with 
magnetism  by  putting  it  into  the  magnetic  field  of  an 
electric  light  or  electric  railway  dynamo.     In  doing  this 
there  is  a  liability  of  reversing  the  poles;  and  therefore, 
after  having  magnetized  the  needle,  place  it  in  the  com- 
pass box  or  rest  it  upon  a  pin  held   in  the   hand,  and 
notice  if  the  poles  have  been  reversed.     If  so,  put  the 
needle  back  into  the  magnetic  field  the  otherend  about. 
A  good  needle  loses  its  magnetism  very  slowly  if  prop- 
erly cared  for  (§  43). 

33.  The  magnetic  axis  of  the  needle  should  coincide  with 
the  axis  of  figure  or  line  connecting  the  two  ends.     An  error 
in  this  respect  affects  only  the  declination  to  be  set  off; 
therefore  it  produces  no  error,  provided  the  declination 
is  determined  by  observing  with  the  instrument  on  a 
true  meridian.     The  amount  of  this  error  could  be  de- 
termined by  inverting  the  needle  on  the  cap  and  reading 
in  both  positions.     "  The  magnetic  axis  is  liable  to  have 
its  position  changed  by  shocks,  in  using  and  transport- 
ing  the   compass.     This   is   especially  true   of   freshly 
magnetized  needles."     For  the  above  reason,  as  well  as 
for  others  which  will  be  discussed  in  Art.  4,  it  is  best  to 
eliminate  the  above  error  by  observing  the  declination 
with  each  instrument  used. 

34.  METAL  OF  COMPASS-BOX.      The  metal  of  the  compass- 
box  should  contain  no  magnetic  substance.     If  the  metal  is 


42  MAGNETIC    COMPASS.  [CHAP.  Ill 

not  pure,  the  effect  on  the  needle  will  be  different  for 
different  positions  of  the  box,  and  consequently  cause 
error.  Iron  is  liable  to  get  mixed  with  the  brass  in 
casting.  To  detect  impure  metal,  set  the  sights  upon 
some  well-defined  object  and  read  the  needle.  Move 
the  vernier  (Chapter  V),  say  10°,  sight  on  the  object, 
and  note  whether  the  needle  has  changed  the  same 
amount.  This  operation  should  be  repeated  all  the 
way  around  the  circle.  If  the  vernier  and  the  needle 
change  like  amounts,  the  metal  of  the  box  is  pure. 
When  reading  the  needle  be  sure  that  nothing  which 
will  affect  it  is  carried  on  the  person.  Watch-chains, 
buttons,  stiffening-wire  in  the  hat  rim,  and  iron  rivets 
in  the  frame  of  the  magnifier  for  reading  the  vernier, 
etc.,  may  produce  error. 

35.  SIGHTS.     The  line  of  sight  should  pass  through  the 
center  of  graduation.     If  this  condition  is  not  satisfied, 
the  angles  will  be  sighted  from  one  point  and  measured 
at  another.     The  error  will  vary  inversely  as  the  length 
of  sight.     There  is  no  probability  that  the  error  will  be 
sufficient  to  affect  the  work  done.     However,  it  could  be 
tested  by  stretching  a  fine  thread  through   the  sights 
and  observing  whether  it  covers  divisions   on  the  limb 
180°  apart. 

36.  ZERO  OF  VERNIER.      The  zero  of  the  vernier  should 
coincide  with  the  line  of  sights.     If  it  does  not,  the  proper 
declination  will  not  be  set  off,  and  an  error  will  be  pro- 
duced in  the  bearings.     To  test  this  condition,  set  the 
vernier  at  zero,  stretch  a  fine  thread  through  the  sights, 
and  observe  whether  it  covers  the  zeros  of  the  gradua- 
tion.    If  it  does  not,  it  will  produce  no  error,  provided 
the  declination  is  determined  with  the  instrument  by  an 
observation  on  a  true  meridian. 

This  condition  should  be  satisfied  for  the  compass  on 
a  transit,  but  there  is  no  rigorous  way  of  making  the 
test.  Hence  it  is  more  important  with  the  transit  than 


ART.  3]  ADJUSTMENT    OF    THE   COMPASS.  43 

with  the  simple  compass  that  the  declination  should  be 
observed  with  each  instrument — compare  the  first  line 
of  Table  I,  page  50,  with  the  other  line. 


ART.  3.     ADJUSTMENT  OF  THE  COMPASS. 

37.  The  study  of  the  adjustments  of  engineering  in- 
struments is  a  very  important  part  of  our  subject,  as  no 
one  is  competent  to  handle  an  instrument  who  is  not 
able  to  determine  when  it  is  in  adjustment,  to  adjust  it 
in  every  particular,  and  to  discuss  the  effect  of  any  error 
of  adjustment  upon  the  work  in  hand.  Instruments 
should  be  examined  frequently,  for  the  adjustments, 
though  properly  made,  are  liable  to  become  deranged. 

Nearly  all  of  the  adjustments  of  engineering  instru- 
ments consist  in  placing  certain  parts  either  perpen- 
dicular or  parallel  to  each  other.  The  usual  method  in 
making  the  various  adjustments  is  that  of  reversions, 
which  doubles  all  errors  and  places  them  on  opposite 
sides,  so  that  if  there  is  no  difference  after  reversal,  there 
is  no  error.  If  there  is  a  difference,  the  mean  of  the  two 
positions  is  the  true  one. 

As  far  as  possible,  the  adjustments  should  be  made 
in  such  a  manner  as  to  be  independent  of  each  other. 
The  student  should  consider  carefully  the  principles 
involved,  and  also  determine  the  effect  upon  subsequent 
work  of  an  error  in  the  adjustment.  The  different 
methods  of  construction  may  modify  the  manner  of 
making  the  adjustments  of  any  instrument,  but  the  same 
general  principles  apply  in  all  cases;  and  hence  the 
great  importance  of  understanding  the  principle  in- 
volved, instead  of  performing  the  adjustments  by  rou- 
tine simply. 

3  8 .  LEVELS.  The  axes  of  the  levels  should  be  perpendicular 
to  the  vertical  axis  of  the  instrument.  The  chief  reason  for 


44  MAGNETIC    COMPASS.  fcHAP.  Ill 

demanding  this  condition  is  that  the  needle  may  play 
freely  when  the  bubbles  are  in  the  center,  whatever  the 
direction  of  the  line  of  sight. 

To  make  this  adjustment,  bring  the  bubble  to  the 
middle  of  the  tube  (any  other  point  would  do  equally 
well,  but  the  center  is  most  convenient)  by  turning  the 
instrument  on  the  ball-and-socket  joint.  Turn  the  in- 
strument half-way  round;  then,  if  the  bubble  does  not 
stand  in  the  middle,  correct  one  half  of  the  difference 
by  means  of  the  screws  at  the  end  of  the  level  tube,  and 
the  other  half  by  turning  the  instrument  on  the  ball- 
and-socket  joint.  This  operation  should  be  repeated 
until  the  bubble  will  remain  stationary  in  the  tube  dur- 
ing a  complete  revolution  of  the  instrument.  If  the 
levels  are  much  in  error,  it  is  best  to  adjust  each  ap- 
proximately before  completing  the  adjustment  of  either. 
Of  course,  before  pronouncing  an  adjustment  in  error, 
it  should  be  carefully  tested. 

39.  The  above  adjustment  of  the  levels  is  sometimes 
erroneously  called  an  adjustment  "  to  cause  the  circle 
to  be  horizontal  in  every  position."     The  adjustment  of 
the  levels  does  not   in  any  way  involve   the   horizontal- 
ity  of  the  plate.     "  Leveling  the  instrument  "  is  really 
bringing  the  vertical  axis  vertical.     It  is   assumed  that 
the  plate  is  perpendicular  to  the  vertical  axis,  and  that 
consequently  when  the   latter  is  vertical    the  former  is 
horizontal.     A  method  of  testing  the  perpendicularity 
of  limb   and  axis  will  be   given  in   the  chapter  on   the 
transit.     For  the  compass   the   above   assumption   will 
produce  no  appreciable  error. 

The  compass  should  not  be  leveled  by  the  needle. 
The  instrument  should  be  leveled  by  the  levels,  and 
then  the  needle  should  be  balanced  by  sliding  the  coil 
of  wire,  which  is  around  the  south  half,  in  or  out. 

40.  SIGHTS.      The  sights  should  be  in  the  plane  of  the  ver- 
tical axis.     This  is  the  rigorous  requirement,  but  for  the 


ART.  3]  ADJUSTMENT    OF    THE    COMPASS.  45 

compass  it  is  sufficiently  exact  to  put  the  sights  in  a 
vertical  plane,  *>.,  in  a  plane  parallel  to  the  plane  of  the 
vertical  axis.  If  the  slits  are  not  vertical,  sighting 
through  the  bottom  of  one  and  the  top  of  the  other 
will  give  a  different  direction  from  that  obtained  by 
sighting  through  the  bottoms  or  tops  of  both. 

To  make  this  adjustment,  bring  the  vertical  axis  ver- 
tical by  the  method  of  the  previous  adjustment.  Re- 
move one  sight,  and  range  two  points,  on  the  side  of 
a  building,  in  the  plane  of  the  remaining  sight.  The 
farther  the  points  are  apart  the  better.  Reverse  the 
instrument  on  the  vertical  axis  and  bring  the  bottom  of 
the  sight  in  range  with  the  lower  point;  if  the  upper 
point  is  then  in  range  with  the  top  of  the  sight,  the 
sight  is  in  a  vertical  plane.  If  the  upper  point  is  not 
in  range,  correct  half  the  error  by  filing  off  one  side  of 
the  bottom  of  the  sight,  or  by  putting  paper  under  the 
other  side.  Adjust  the  other  sight  in  the  same  way. 
The  plane  of  the  sights  is  now  vertical,  but  it  may  not 
coincide  with  the  plane  of  the  vertical  axis.  The  above 
method  of  correcting  the  error  is  best,  but  in  case  of 
necessity  it  may  be  done  by  holding  both  sights  and 
bending  the  plate  as  needed.  With  fair  use  the  sights 
should  not  get  out  of  adjustment. 

41.  NEEDLE,  a.  The  ends  of  the  needle  and  the  point  of 
the  pivot  should  be  in  the  same  horizontal  plane.  This  con- 
dition is  required  so  that  the  ends  of  the  needle  may 
remain  stationary  even  though  the  needle  may  roll  or 
quiver  on  the  pivot.  To  make  this  adjustment,  bend 
the  ends  of  the  needle  up  until  they  remain  steady  even 
though  the  middle  of  the  needle  may  swing. 

b.  The  ends  of  the  needle  and  its  center  should  be  in  the 
same  vertical  plane.  If  this  condition  is  not  satisfied,  the 
readings  of  the  two  ends  of  the  needle  will  not  agree. 
To  test  it,  read  both  ends,  and  revolve  the  box  until  the 
north  end  reads  what  the  south  end  did.  If  there  is  a 


46  MAGNETIC    COMPASS.  [CHAP.  Ill 

difference  between  the  second  reading  of  the  south  end 
and  the  first  reading  of  the  north  end,  correct  half  of 
the  difference  by  bending  the  needle. 

42.  CENTER-PIN.      The  pivot  on  which  the  needle  swings 
should  be  in  the  center  of  the  graduated  circle.     If  the  pivot 
is  not  in  the  center,  either  the  needle  will   strike  the 
graduation  and  not  turn  freely,    or  the  end  will  be  so 
far  from  the  graduation   that  it  can  not  be  read  pre- 
cisely.     Furthermore,  if  this  condition  is  not  satisfied, 
a  bearing  read  from  the   north  end  of  the  needle  will 
not  agree  with  the  value  obtained  from  the  south  end. 
To  test  this  adjustment  read  one  end  of  the  needle  in 
four  positions  90°  apart  ;  if  the  readings  of  the  other 
end  differ  by  90°,  the  pivot  is    in    the   center.     If   the 
readings  of  the  second  end  do  not  differ  by  90°,  read 
both  ends  of  the  needle  again,  move  the  box  until  one 
end  has  passed  over  90°  and  then  bend  the  pivot  until 
the  other  end   shall  have   passed  over  90°.     Turn   the 
box  about  90°  and  repeat  the  last  operation.     Test  the 
work    by  trying   points    half-way    between    those    pre- 
viously used. 

This  adjustment  can  be  made,  but  less  elegantly,  by 
using  half  the  length  of  the  needle  to  measure  the  dis- 
tance from  the  pivot  to  the  graduation. 

This  adjustment  should  be  examined  every  time  the 
pivot  is  taken  out.  Notice  that  the  adjustment  of  the 
needle  and  of  the  center-pin  are  entirely  independent, 
text-books  to  the  contrary  notwithstanding. 

ART.  4.     USING  THE  COMPASS. 

43.  CAKE.      It  is   quite  important   that  the  engineer 
should  understand  the  means  by  which  he  can  prolong 
the  usefulness  of  his  instrument.     In   the  compass  the 
main  thing  is   to  avoid  dulling  or  breaking  off  the  fine 
point  of  the  pivot;    consequently  never  jolt   or  carry 


ART.  4]  USING    THE    COMPASS.  47 

the  compass  without  being  sure  that  the  needle  is 
lifted  off  the  pin.  Remember  that  the  harder  and 
more  perfect  the  point,  the  more  liable  it  is  to  injury. 
In  lowering  the  needle  do  not  let  it  fall  upon  the  pivot. 
To  prevent  unnecessary  wear,  check  the  vibrations  of 
the  needle  on  letting  it  down,  by  lifting  it  off  the 
point  a  little;  or,  better,  turn  the  needle  in  about  the 
proper  direction  before  letting  it  down. 

When  the  pivot  becomes  dulled  it  can  be  sharpened 
on  an  oil-stone.  Care  should  be  taken  to  obtain  a  coni- 
cal and  not  a  pyramidal  point.  The  sharpness  of  a 
needle  is  easily  ascertained  by  sliding  the  thumb-nail 
over  the  point,  at  an  angle  of  about  30°  to  it.  If  the 
point  sticks  and  holds  the  nail,  it  is  sharp;  if  it  glides 
upon  it,  it  is  dull.  Unnecessary  grinding  of  the  pivot 
should  be  avoided,  for  if  it  becomes  much  shortened, 
the  ends  of  the  needle  will  come  below  the  graduation, 
thereby  producing  parallax  in  reading.  This  parallax 
could  be  avoided  by  bending  up  the  ends  of  the  needle, 
but  this  would  destroy  the  condition  that  the  two  ends 
of  the  needle  and  the  pivot  should  be  in  the  same  hori- 
zontal plane  (see  a,  §  41). 

Never  allow  the  needle  to  be  played  with  by  at- 
tracting it  with  a  knife,  piece  of  iron,  etc.;  for  every 
passage  of  a  piece  of  steel  or  iron  removes  a  portion  of 
the  magnetism.  When  the  compass  is  not  in  use,  it  is 
best  to  let  the  needle  assume  its  normal  position  in  the 
magnetic  meridian,  for  in  this  position  it  will  longer  re- 
tain and  even  increase  its  polarity.  After  the  needle  has 
assumed  this  position  it  should  be  raised  against  the 
glass. 

44.  PRACTICAL  HINTS.  If  the  needle  is  sluggish  in  its 
movements  and  settles  quickly,  either  it  has  lost  its 
magnetic  force  or  it  has  a  blunt  pivot;  and  in  either 
case  it  is  likely  to  settle  considerably  out  of  its  true 
position.  The  longer  a  needle  is  in  settling,  the  more 


48  MAGNETIC    COMPASS.  [CHAP.  Ill 

accurate  will  be  its  final  position.  It  can  be  quickly 
brought  very  near  to  its  true  position  by  checking  its 
motion  by  means  of  the  lifting  screw;  but  in  its  final 
settlement  it  must  be  left  free. 

The  glass  cover  may  become  electrified  from  friction 
and  attract  the  needle.  This  electricity  can  be  dis- 
charged by  touching  the  glass  with  a  wet  finger,  or  by 
breathing  upon  it. 

45.  SOURCES  OF  ERROR.*     The  errors  of  compass  work 
may  be  classified  as  (i)  local  attraction,  (2)  instrumental 
errors,  and  (3)  observational  errors. 

46.  Local  Attraction.     A  common   method  of  taking 
the  bearings  with  the  compass  is  to  set  it  at  each  cor- 
ner or  station  in  succession,  and  take  the  forward  bear- 
ing.    If  there  is  any  local  attraction  of  the  needle,  i.e., 
if  the  needle  is  deflected  from  its  normal  direction  by 
iron,  etc.,  the  bearings  will  be  incorrect.     A  method  of 
eliminating  all  local  attraction  is  described  in  Appendix 
I,  and  hence  this  source  of  error  need  not  be  discussed 
here. 

47.  Instrumental  Errors.     Errors  of  this  class  are  due 
either  to  imperfect  adjustment  of  the  instrument  or  to 
sluggishness  of  the  needle.     It  is  a  good  rule  to  adjust 
an  instrument  carefully  in  all  particulars,  and  then  use 
it  in   such  a  way  as  to  eliminate  any  residual  errors  of 
adjustment  ;  or,  in  other  words,  adjust  it  carefully  and 
then  use  it  as  though  it  were  not  in  adjustment. 

Guard  against  errors  of  coincidence  of  magnetic  and 
geometrical  axes  of  the  needle  (§  33),  and  also  against 
errors  in  straightness  of  needle  (§  41),  by  reading  al- 
ways the  same  end  of  the  needle. 

48.  In  work  involving    the  magnetic    declination,  to 
guard  against  the  possibility  (i)  that  the  magnetic  axis 
of  the  needle  may  not  coincide  with  its  geometric  axis, 

*  See  discussion  ot  Cumulative  vs.  Compensating  Error,  §  18, 


ART.  4]  USING    THE    COMPASS.  49 

(2)  that  the  zero  of  the  vernier  may  not  coincide  with 
the  line  of  sight,  and  (3)  that  the  needle  is  not  straight, 
determine  the  declination  by  setting  the  compass  upon 
a  true  meridian,  sighting  along  it,  and  then  moving  the 
vernier  until  the  needle  reads  zero.  This  also  provides 
against  errors  in  charts  or  tables  giving  the  declination, 
and  also  against  changes  in  the  declination  since  the 
chart  was  made.  This  observation  should  be  made 
about  10  A.M.  or  6  P.M.,  as  then  the  effect  of  the  daily 
variation  is  generally  nearly  zero. 

It  is  very  important  that  the  three  sources  of  error 
just  mentioned  should  be  eliminated.  For  one  or  the 
other  or  all  of  these  reasons,  the  bearing  of  a  line  as  read 
from  several  instruments  at  the  same  time  and  place, 
by  the  same  person,  will  often  differ  considerably.  In 
one  instance,*  the  magnetic  declination  as  read  from 
five  instruments  by  four  men  at  the  same  time  and 
place  differed  20  minutes.  As  nearly  as  can  be  deter- 
mined from  so  few  observations,  the  probable  error  (see 
Appendix  III)  of  a  single  determination  of  the  declina- 
tion is  5^  minutes.  In  another  instance,*  the  readings 
of  four  men  from  four  instruments  read  at  the  same 
time  and  place  differed  31  minutes,  with  a  probable 
error  for  each  observation  of  9.1  minutes.  The  follow- 
ing is  from  a  prominent  instrument  maker  :f  "I  made 
six  needles  as  near  alike  as  I  could.  I  then  set  up  a 
compass  and  directed  it  toward  a  certain  fixed  object. 
Then  I  placed  these  needles,  made  at  the  same  time  of 
the  same  kind  of  material,  on  the  center-pin  one  after 
another.  Three  of  these  gave  the  same  reading,  but  the 
other  three  varied  from  5  to  10  minutes.  The  result 
was  indeed  surprising,  for  I  had  taken  great  pains  to 
have  the  needles  all  alike." 


*  Report  of  the  Ohio  Society  of  Surveyors  and  Engineers  for  1884,  p.  71. 
fid.,  p.  74. 


5° 


MAGNETIC    COMPASS. 


[CHAP,  in 


In  the  course  of  ordinary  class  instruction,  the  author 
had  his  class  in  land-surveying,  consisting  of  eleven 
members,  determine  the  magnetic  declination  by  ob- 
serving upon  a  true  meridian  with  six  instruments  at 
the  same  time,  each  man  observing  with  each  instrument. 
At  the  time  of  making  the  reading  no  one  knew  what 
the  others  had  read.  The  instruments  were  in  good 
adjustment  in  every  particular.  The  following  table 
shows  the  mean  declination  for  each  instrument,  and 
also  the  probable  error*  of  observation. 

TABLE  I. 

DIFFERENCE  IN  MAGNETIC  DECLINATIONS  OBTAINED  WITH  DIFFER- 
ENT INSTRUMENTS. 


No. 

Instrument. 

Length  of 
Needle. 

Mean 
Declination. 

Probable 
Error  of  a 
Single 
Observation. 

I 

Black  transit  

5  inch 

5°  42' 

I  .5 

2 

Yellow     "           

c      ' 

4°  27' 

2  .8 

q 

Mining    "       

3     ' 

4°  33' 

I  .6 

Black  compass 

6     ' 

4°     4O 

i  .6 

5" 

Yellow      " 

6     ' 

4°  m' 

I  .2 

6 

Old            " 

5     « 

4°  37' 

l'.2 

4°   484-' 

i'.7 

Since  each  value  of  the  declination  in  the  above  table 
is  the  result  of  eleven  observations,  the  error  of  obser- 
vation is  practically  eliminated,  and  hence  the  difference 
in  the  declinations  is  due  almost  solely  to  a  difference 
in  the  instruments.  The  above  results  show  that  the 
declination  determined  by  a  single  observation  has  a 
probable  error  (§  2  of  Appendix  III)  of  1.7  minutes  due 

*  Since  the  readings  were  made,  as  a  rule,  only  to  the  nearest  5  minutes,  it 
is  not  strictly  correct  to  compute  the  probable  error  of  such  results.  The  prob- 
able errors  are  given  in  this  case  to  show  that  the  difference  in  the  declination 
was  due  to  the  instruments,  and  not  to  the  reading. 


£RT.  4]  USING    THE    COMPASS.  5  I 

to  the  observer  and  13.4  minutes  due  to  the  instrument; 
or  a  total  probable  error  of  13.5  minutes.  In  other 
words,  if  both  instruments  are  in  good  working  condi- 
tion and  skilfully  used,  we  may  expect  a  difference  in 
the  [bearing  of  the  line  due  to  differences  in  the  in--" 
struments,  of  13.4  minutes  (21  feet  in  a  mile),  as 
read  by  two  instruments  at  the  same  time  and  place. 
Notice  that  the  greatest  difference  between  the  declina- 
tions is  i°  15'  (120  feet  in  a  mile).  This  source  of  error 
is  generally  disregarded,  and  frequently  it  is  impossible 
to  do  otherwise;  but  it  should  be  continually  borne  in 
mind  as  a  possible  source  of  error. 

49.  Observational  Errors.  These  may  be  divided  into 
(i)  errors  in  sighting  and  (2)  errors  in  reading.  There 
is  a  possibility  of  error  owing  to  the  compass  or  flag- 
pole's  not  being  set  at  exactly  the  right  point;  but  this 
can  only  occur  on  short  sights,  and  with  reasonable  care 
will  not  occur  at  all. 

1.  To  eliminate  any  residual  errors  due  to  the  plate's 
not  being  level  (§  38)  or  to  the  sights'  not  being  perpen- 
dicular to  the  plate  (§  40),  sight  through  the  top  or  bot- 
tom of  both  sights.     To  insure  the  slits'  being  vertical, 
give  the  most  care  to  the  bubble  perpendicular  to  the 
line  of  sight.     Care  must  be  taken  that  the  flag-pole  is 
seen  through  both   slits,  and  not  through  one  slit  and 
one  hole.     The  flag-pole  should  be  set  vertical ;  but,  as 
a  precaution,  sight  as  low  on  it  as  possible. 

2.  After  the  needle  has  come  to  rest    it   should  be 
tapped  gently  to  destroy  the  effect  of  any  adhesion  to 
the  pivot.     In  reading  the  needle  beware  of  magnetic 
substances  on  the  person  as  wire  in  the  hat-brim,  watch- 
chains,  rivets  in  the  handle  of  the  magnifying-glass,  etc., 
which  may  affect  the  needle.     The  most  common  errors 
made  in  practice  are  such  as  reading  28°  for  32°,  30^° 
for  29^°,  etc.;  and  reading  N.  for  S.,  etc.     The  remedy 
is  obvious. 


52  MAGNETIC    COMPASS.  [CHAP.   Ill 

The  common  practice  is  to  read  the  bearings  to  the 
nearest  quarter-degree.  The  accuracy  of  the  work  can 
be  appreciably  increased  by  estimating  the  fraction  of 
a  degree  to  the  nearest  five  minutes.  Since  compasses 
are  ordinarily  graduated  to  half-degrees,  this  necessi- 
tates the  estimation  of  sixths  of  a  division.  '  With  a 
little  thoughtful  attention  this  can  be  done  with  con- 
siderable precision,  and  the  increased  accuracy  is  well 
worth  the  extra  time  required. 

It  is  sometimes  recommended  that  the  vernier  be 
used  to  read  the  fractions  of  a  degree  ;  but  a  trial  will 
show  that  this  is  no  advantage.  The  reasons  are  obvious. 
It  has  also  been  proposed  to  place  a  light  paper  vernier 
on  the  south  end  of  the  needle.  Concerning  all  such 
devices  remember  that  if  the  needle  were  read  without 
any  error  at  all,  the  magnetic  compass  would  not  be  an 
instrument  of  any  considerable  precision. 

50.  LIMITS  OF  PRECISION.     With  a  compass  graduated 
to   half-degrees,  the   angles   can    be   estimated    to    the 
nearest  5  minutes,  in  which  case  the  maximum  error  of 
the  bearing  should  not  exceed  10  minutes.     The  average 
error  of  a  bearing  by  the  author's  students,  as  deduced 
from   the  errors  of  closing   the  angles   around  a  field 
(§  8  of  Appendix  I),  the  average  length  of  sight  being 
about  300  feet,  is  3  minutes.*     The  average  error  for  ten 
selected  men,  using  a  highly  magnetized  needle  and  a 
sharp  pivot,  was  2.5  minutes.     The  same  men  read,  for 
the  purpose  of  this  record,  the  bearing  of  a  flag-pole  at 
100,  200,  and  300  paces,  with  average  errors  of  4,  5,  and 
5.2  minutes,  respectively,  the  sun  and  wind  being  in  the 
observer's  face  while  sighting. 

51.  The  error  of  an  area  found  by  compass  surveying 
is  made  up   of  the  errors  of  chaining,  of  reading  the 


*  The  error  per  sight  is  equal  to  the  total  angular  error  in  closing  a  field, 
divided  by  the  square  root  of  twice  the  number  of  sides, 


ART.   4]  USING    THE    COMPASS.  53 

bearings,  and  of  the  computations.  With  proper  care, 
the  error  of  chaining  should  be  much  less  than  the  error 
of  the  bearings.  Since  i  in  57.3  corresponds  to  i°,  the 
above  maximum  error  of  10'  in  reading  an  angle  corre- 
sponds to  an  error  of  i  in  344  ;  and  the  average  error  of 
3',  as  above,  corresponds  to  i  in  1,146,  which  is  a  greater 
error  than  that  of  chaining  under  the  same  conditions 
(see  fourth  paragraph  of  §  22).  Since  the  computations 
are  self-checking  at  nearly  every  step,  there  is  little 
probability  of  any  material  undetected  error  in  this  part 
of  the  work  ;  and  as  the  work  may  be  carried  to  any 
desired  number  of  decimals,  the  inaccuracies  of  the 
computations  can  be  made  much  less  than  those  of  the 
field-work.  We  conclude,  therefore,  that  the  accuracy 
of  the  area  depends  mainly  upon  the  accuracy  of  read- 
ing the  needle. 

The  average  error  in  area  of  eighty  problems  solved 
by  thirty-two  (the  best  of  thirty-six)  of  the  author's 
students  in  ordinary  class-work*  was  i  in  1,530.! 

52.  BALANCING  THE  LATITUDES  AND  DEPARTURES.    An 

answer  can  now  be  given  to  the  following  question, 
which  is  frequently  asked  :  How  great  a  difference 
between  the  sums  of  the  +  and  —  latitudes  and  de- 
partures is  admissible  ? 

Let  C  represent  the  linear  error  of  closure  due  to  the 
chaining,  P  the  perimeter  of  the  field,  d  the  distance 
in  which  the  error  of  chaining  is  a  unit ;  then 


*  See  foot-note,  p.  28. 

t  For  the  sake  of  comparisons,  it  may  be  interesting  to  know  that  under 
the  same  conditions  the  error  of  areas  obtained  with  the  chain  alone  was  i  in 
1,520;  and  with  a  home-made  plane  table  (§  172)  the  error  was  by  radiation 
(§  182)  i  in  586,  by  traversing  (§  184)  i  in  826,  and  by  radio-progression  (§  187) 
i  in  1,111. 


54  MAGNETIC    COMPASS.  [CHAP.    Ill 

Let  A  represent  the  lineal  error  due  to  the  measure- 
ment of  the  angles  ;  and  a  the  angular  error  of  meas- 
uring the  angles,  i.e.,  the  difference  between  the  last 
fore-sight  and  the  first  back-sight,*  in  minutes.  It 
can  not  be  known  how  this  error  occurred,  whether  all 
in  one  sight,  or  equally  among  the  sides,  or  among  the 
sides  in  proportion  to  their  length  ;  but  as  the  last  as- 
sumption is  most  probable,  and  also  as  it  gives  the 
largest  linear  error  in  closing,  we  will  assume  the  error 
a  to  have  occurred  among  the  sides  in  proportion  to 
their  lengths.  Hence  to  reduce  a  to  its  linear  equiva- 
lent, we  must  multiply  it  by  the  length  of  the  perimeter 
(=  P)  and  divide  it  by  the  distance  at  which  a  unit 
subtends  an  angle  of  one  minute,  or 

A  =  a  --—  —  — —  ,  nearly.     ...     (2) 
3,438       10,000' 

Let  E  =  the  total  error  due  to  chaining  and  to 
measuring  the  angles.  Notice  that  the  error  E  is  the 
hypotenuse  of  a  right-angled  triangle  of  which  the 
differences  of  the  latitudes  and  departures  are  the  other 
sides.  Hence,  if  L  =  the  difference  of  the  -j-  and  — 
latitudes,  and  D  =  the  difference  between  the  -}- 
departures,  E—  V Dl  -f  L\ 

By  the  theory  of  probabilities  we  know  that 


-+ — —  V  nearly- 

1 2, 000,0007 


See  §  8  of  Appendix  I. 


ART.  4]  USING    THE    COMPASS.  55 

I    Equating  these  two  values  of  £,  we  get 


12000000 


As  a  rule,  equation  (4)  is  the  one  to  be  used  in  prac- 
tice ;  but  we  may  simplify  the  matter  a  little  further  by 
finding  a  relation  between  D  and  L.  Assume  that  the 
sum  of  the  latitude  =  n  times  the  sum  of  the  departures. 
n  is  easily  determined  from  the  computations,  or  it  can 
be  estimated  in  the  field  or  from  the  plat  with  sufficient 
accuracy.  Then,  from  the  theory  of  probabilities, 
L  =  D  Vn.  Equation  (4)  then  becomes 


(s) 


12,000,0007 


53.  To  illustrate  the  method  of  applying  the  preced- 
ing formulas,  let  us  assume  that  in  surveying  a  field 
whose  perimeter  is  10  chains  the  difference  between 
the  first  back-sight  and  the  last  fore-sight  was  10 
minutes.  We  will  also  assume  that  the  conditions 
were  such  that  we  might  expect  an  error  in  chaining  of 
i  in  2,000.  Equation  (4)  then  becomes 


=  — 1 —  =  .00086.     .     .    (6) 

40,000          1,200 

If  the  field  is  twice  as  long  north  and  south  as  east 
and  west,  then  n  =  2.  Substituting  in  equation  (6)  and 
reducing,  we  get  3  D*  =  0.00086  chains  ;  therefore, 
Z>  =  0.017  chains  =  1.7  links.  This  shows  that  the 


56  MAGNETIC   COMPASS.  fcttAP.  Ill 

error  in  the  departures  should  be  about  1.7  links.  The 
error  in  the  latitude  should  be  1.7  Vn  =  1.7  ^2  =  2.4 
links.  Results  much  greater  than  these  show  an  error 
in  the  work  other  than  the  usual  inaccuracy. 

Finally,  knowing  the  error  of  closing  the  angles  and 
the  errors  in  balancing  the  latitudes  and  departures,  we 
may  reverse  the  problem  and  compute  the  error  of 
chaining.  To  do  this,  notice  that  all  quantities  in  equa- 
tion (4)  would  then  be  known  except  d>  which  could 
then  be  computed. 


CHAPTER  IV. 
SOLAR  COMPASS.* 

54.  THE  solar  compassf  is  the  result  of  an  attempt  to 
overcome  the  defects  of  the  magnetic  compass  due  to 
the  variation  in  the  magnetic   declination.     The    solar 
compass   determines   lines   with   reference   to   the    sun; 
and  hence  the  bearings  are  independent  of  any  changes 
of  the  magnetic  needle.     The  solar  compass  consists  of 
an  ordinary  magnetic  compass,  to  which  is  attached  an 
apparatus  for  sighting  at  the  sun,  briefly  called  the  solar 
apparatus. 

55.  PRINCIPLE  OF  THE  SOLAR  APPARATUS.    In  its  most 

elementary  form  the  solar  apparatus  consists  of  a  right 
line  parallel  to  the  axis  of  the  earth,  to  which  is 
pivoted  another  right  line  which  is  free  to  move  only 
in  the  plane  of  the  first  line.  For  convenience  of 
explanation,  call  the  first  line  the  polar  axis,  and  the 
second  the  solar  sight-line. 

If  the  polar  axis  is  parallel  to  the  earth's  axis,  and 
the  solar  sight-line  is  directed  towards  the  sun,  then,  as 
the  solar  sight-line  is  revolved  about  the  polar  axis,  it 
will  trace  on  the  sky  the  diurnal  path  of  the  sun.  But, 
if  the  polar  axis  is  not  parallel  to  the  axis  of  the  earth 
and  the  angle  between  the  axis  and  the  sight  line  re- 
mains the  same  as  before,  it  will  not  be  possible  to 

*  For  a  discussion  of  the  Solar  Transit,  see  Chapter  VIII.    . 
t  Invented  by  Wm.  A.  Burt,  a  U.  S.  public-land  surveyor  of  Michigan,  in 
1836. 

57 


58  SOLAR    COMPASS.  [CHAP.  IV 

make  the  sight  line  point  to  the  sun.*  In  other  words,  if 
the  polar  axis  of  the  solar  apparatus  is  parallel  to  the 
axis  of  the  earth,  and  if  the  sight  line  makes  an  angle 
with  the  perpendicular  to  the  polar  axis  equal  to  the 
declination  of  the  sun,  then  the  sight  line  can  be 
brought  to  bear  upon  the  sun  only  when  the  plane  of  the 
two  lines  coincides  with  the  true  meridian.  This  is  the  fun- 
damental principle  of  all  solar  apparatuses,  however 
much  they  may  differ  in  detail. 

The  solar  apparatus,  then,  consists  (i)  of  a  device  for 
making  the  polar  axis  parallel  to  the  axis  of  the  earth, 
i.e.,  for  giving  the  polar  axis  an  angle  of  elevation 
equal  to  the  latitude  of  the  place  of  observation;  (2)  of 
a  device  for  setting  off  an  angle,  between  the  perpen- 
dicular to  the  polar  axis  and  the  solar  sight-line,  equal 
to  the  declination  of  the  sun;  and  (3)  of  some  device 
for  revolving  the  solar  sight-line  about  the  polar  axis. 
If  the  solar  apparatus  is  attached  to  a  common  compass 
or  transit  in  such  a  manner  that  the  plane  of  the  polar 
axis  is  parallel  to  the  plane  of  the  terrestrial  sight-line 
of  the  instrument,  then  when  the  latitude  of  the  place 
of  observation  and  the  declination  of  the  sun  are  cor- 
rectly set  off,  and  the  solar  sight-line  is  directed  to  the 
sun,  the  terrestrial  sight-line  will  indicate  a  true  me- 
ridian. 


ART.  1.     CONSTRUCTION  OF  THE  SOLAR  COMPASS. 

56.  The  usual  form  of  the  solar  compass  is  shown  in 
Fig.  8.  It  consists  essentially  of  three  arcs  of  circles 
by  which  the  latitude  of  a  place,  the  declination  of  the 
sun,  and  the  hour  of  the  day  may  be  set  off.  These 

*  Strictly,  if  these  conditions  are  not  fulfilled  the  solar  sight-line  can  be 
made  to  point  toward  the  sun  but  once,  or  at  most  only  twice,  during  the 
twenty-four  hours,  and  then  only  for  an  instant. 


ART.    l]     CONSTRUCTION    OF    THE    SOLAR   COMPASS.  59 

arcs  are  designated  in  the  cut  by  the  letters  a,  b,  and  c, 
respectively. 

The   latitude  arc  a  has  its  center  of  motion  in  two 


FIG.  8.— SOLAR  COMPASS. 


pivots,  one  of  which  is  shown  in  Fig.  8,  at  d.  The  lati- 
tude arc  is  moved  either  up  or  down  within  a  hollow 
arc,  seen  in  the  cut,  by  a  tangent  screw  at  /,  and  may 
be  fastened  in  any  position  by  a  clamp  screw  e.  The 


60  SOLAR  COMPASS.  [CHAP,  iv 

vernier  of  the  latitude  arc  usually  reads  to  single  min- 
utes. 

The  declination  arc  b  has  a  range  of  about  25°,  and 
is  read  by  a  vernier  to  minutes.  The  arm  carrying 
the  vernier  is  moved  over  the  surface  of  the  declination 
arc,  and  its  vernier  set  to  any  reading,  by  turning  the 
head  of  the  tangent  screw  k.  The  vernier  is  clamped  in 
any  position  by  a  screw,  not  seen  in  Fig.  8. 

At  each  end  of  the  collimation  arm  h  is  a  rectangular 
block  of  brass.  In  the  upper  of  these  blocks  is  set  a 
small  convex  lens,  having  its  focus  on  the  surface  of  a 
little  silver  or  ivory  plate  fastened  to  the  inside  of  the 
opposite  block.  On  the  surface  of  the  plate  are  two 
pairs  of  lines  intersecting  each  other  at  right  angles. 
The  interval  between  the  lines  of  each  pair  is  just 
sufficient  to  include  the  circular  image  of  the  sun  as 
formed  by  the  solar  lens. 

Each  end  of  the  arm  h  has  both  a  lens  and  a  silver  or 
ivory  plate.  As  shown  in  Fig.  8  the  compass  is  set  for 
south  declination;  and  when  the  sun  has  north  declina- 
tion, the  declination  arc  and  the  solar  line  of  sight  are 
turned  180°  on  the  polar  axis,  and  the  observation  is 
made  with  the  lens  and  plate  on  the  end  of  the  arm  // 
opposite,  respectively,  to  those  used  for  a  south  declina- 
tion of  the  sun.  Thus  there  are  really  two  solar  sight- 
lines — one  for  north  and  one  for  south  declination. 

57.  When  the  instrument  is  leveled,  and  the  co-lati- 
tude is  set  off  on  the  arc  a,  and  the  declination  of  the 
sun*  is  set  off  on  the  arc  bt  if  the  whole  instrument  is 
revolved  about  its  polar  axis,/,  until  the  image  of  the 
sun  formed  by  the  lens  in  the  upper  end  of  the  collima- 
tion arm,  ^,  falls  in  the  middle  of  the  lines  on  the  plate 
on  the  lower  end  of  the  collimation  arm,  then  the  terres- 
trial line  of  sight  of  the  instrument  will  be  on  a  true  meridian. 

*  After  being  corrected  for  refraction  (§  159). 


ART.   2]       ADJUSTMENT    OF    THE    SOLAR    COMPASS.  6l 

The  angle  between  any  course  and  the  true  meridian 
may  be  read  by  the  graduation  on  the  horizontal  plate 
of  the  instrument. 

58.  As  solar  work  can  be  performed  only  during  clear 
weather,  the  instrument  is  provided  with  an  ordinary 
magnetic  needle — shown  at  n  in   Fig.  8 — by  which    to 
run  lines  in  cloudy  weather.     The  needle  may  be  used 
also  to  determine  the  magnetic  declination.     Instead  of 
sights  the  solar  compass  is  sometimes  provided  with  a 
telescope;  but  this  addition  is  of  little,  if  any,  advan- 
tage. 

ART.  2.    ADJUSTMENT  OF  THE  SOLAR  COMPASS. 

59.  The  following  adjustments  of  the  solar  compass 
must  be  carefully  attended  to:     i.  The    plane  of   the 
plate  bubbles  should  be  perpendicular  to  the  vertical 
axis.     2.  The  two  solar  sight-lines  should  be  parallel. 
3.  The  declination  arc  should  read  zero  when  the  solar 
sight-line  is  perpendicular  to  the    polar  axis.     4.  The 
latitude  arc  should   read  the  latitude  when   the  polar 
axis  is  parallel  to  the  axis  of  the  earth.*     5.  The  terres- 
trial line  of  sight  and  the  solar  line  of  sight  should  lie 
in  the  same  vertical  plane. 

The  method  of  making  these  adjustments  will  not  be 
described  here,  for  the  following  reasons:  i.  It  will  not 
be  difficult  for  any  one  familiar  with  the  methods  of 
adjusting  engineers'  field-instruments  to  devise  ways  of 
making  these  adjustments.  2.  These  adjustments  are 
similar  to  those  of  the  solar  transit  to  be  described 
presently.  3.  On  account  of  the  defects  of  the  solar 
compass  it  is  better  to  use  either  the  magnetic  compass 
(Chapter  III)  or  the  solar  transit  (Chapter  VIII). 

*  If,  however,  the  value  of  the  latitude  used  with  the  instrument  be  that 
obtained  by  it,  then  no  attention  need  be  paid  to  this  adjustment.  It  is  im- 
portant only  when  the  true  latitude  is  used  in  finding  the  meridian,  or  when 
the  true  latitude  of  the  place  is  to  be  found  by  the  instrument. 


62  SOLAR   COMPASS.  [CHAP.  IV 


ART.  3.     USING  THE  SOLAR  COMPASS. 

60.  MERITS   AND   DEFECTS   OF   THE   SOLAR    COMPASS. 

Merits,  i.  The  chief  merit  claimed  for  the  solar  com- 
pass is  that  lines  run  by  it  are  not  affected  by  vari- 
ations of  the  magnetic  needle  ;  but  the  common  com- 
pass can  be  used  (Appendix  I)  so  as  to  be  absolutely 
independent  of  variations  of  the  needle,  and  hence  this 
claim  has  no  force.  2.  A  second  merit  claimed  is  that 
the  solar  compass  enables  the  surveyor  to  obtain  a  true 
meridian  very  easily  ;  but  as  there  are  several  other 
ways  of  finding  a  true  meridian,  this  advantage  is  not 
very  great.  3.  It  is  also  claimed  that  the  solar  compass 
is  peculiarly  valuable  in  the  U.  S.  public-land  surveys 
in  running  parallels  of  latitude  ;  but  as  the  transit  is 
more  accurate  and  more  convenient  for  this  purpose, 
this  point  is  not  well  taken. 

61.  Defects.     The   solar   compass   has   the   following 
defects:  i.  It  can  be  used  only  when  the  sun  is  shining. 
At  other  times  it  can  be  no  more  accurate  than  the  mag- 
netic compass,  while  it  is  much  heavier  and  more  com- 
plicated.    2.  Owing  to  its  many  adjustments   and  the 
difficulty  of  making  and  keeping  them,  the  solar  com- 
pass is  not   an  exact  instrument  even  in  clear  weath- 
er.    "  A  state  boundary-line  run  with  the  solar  compass 
and  intended  to  be  straight  was  found   to  deviate  ten 
minutes  when  re-run   with  a  transit."     3.  To  use  the 
solar  compass,  the  engineer  must  know  his  latitude,  the 
declination  of  the  sun  for  each  hour  of  the  day,  the  time 
of  day,  and   the   refraction   for  all  altitudes;  and   the 
inevitable  inaccuracies,  not  to  consider  large  errors,  in 
these  data  render  work  done  with   the  solar  compass 
unreliable.     For  example,  an  error  of  one  minute  in  the 
declination  may  cause  an  error  of  twelve  minutes  in  the 
direction  of  the  supposed  meridian. 


ART.  3]  USING    THE    SOLAR    COMPASS.  63 

62.  HISTORY.  The  solar  compass  was  invented  by 
Wm.  A.  Burt,  a  deputy  U.  S.  land-surveyor  in  the  Lake 
Superior  mineral  region,  in  1836 — about  the  time  of  the 
introduction  of  the  American  transit.  Previously  the 
magnetic  compass  and  the  heavy  and  inconvenient 
English  theodolite  were  the  only  angle  instruments  in 
general  use  in  this  country.  The  solar  compass  was 
much  lighter  and  more  convenient  than  the  theodolite, 
and  more  accurate  than  the  magnetic  compass  as  «*^n 
made  and  used.  This  probably  accounts  for  the  favor 
with  which  the  solar  compass  has  been  received  in  the 
past.  In  popular  estimation  it  is  an  instrument  of  great 
precision,  while  in  reality  it  is  little,  if  any,  more  accu- 
rate than  the  magnetic  compass.  It  is  certainly  neither 
so  accurate  nor  capable  of  so  great  a  variety  of  work 
as  the  engineer's  transit,  which  is  properly  employed 
where  formerly  the  solar  compass  was  used. 

The  solar  compass  is  certainly  a  very  ingenious  in- 
strument, and  reflects  more  credit  upon  the  inventor  than 
upon  the  modern  user.  The  form  shown  in  Fig.  8  is 
the  one  in  use  at  the  present  time,  and  is  essentially  the 
form  originally  given  to  it  by  the  inventor. 

We  will  not  further  discuss  this  instrument,  but  refer 
the  reader  to  Burt's  "  Key  to  the  Solar  Compass,"  in 
which  the  inventor  fully  describes  the  adjustment  and 
use  of  his  instrument. 


CHAPTER  V. 


VERNIERS. 

63.  A  VERNIER*  is  a  short  scale  movable  by  the  side 
or  a  longer  scale,  by  which  subdivisions  of  the  longer 
scale  may  be  measured.     The  scale  to  be  subdivided  is 
called  a   limb.     A    division    of   the  vernier   is    a    little 
shorter  or  a  little  longer  than  a  division  of  the  main 
scale  or  limb.     This  small  difference  is   the  unit  of  the 
space  measured  by  the  vernier. 

64.  PRINCIPLES.      A  vernier  may  be  constructed  by 
taking  a  length  equal   to  any  number  of  parts  of  the 

limb,  and  dividing  it  into  a  number  of 
equal  parts,  one  more  or  one  less  than 
the  number  into  which  the  same  length 
on  the  limb  is  divided.  For  example, 
the  limb  shown  in  Fig.  9  is  a  scale  of 
inches  divided  into  tenths,  and  the  di- 
visions of  the  vernier  are  of  such  a 
length  that  ten  spaces  on  the  vernier 
are  equal  to  nine  on  the  limb  or  main 
scale.  Therefore  each  space  on  the 
vernier  is  equal  to  o.i  of  0.9,  or  0.09,  of 
an  inch;  that  is,  each  space  on  the 
vernier  is  o.oi  of  an  inch  shorter  than 
a  space  on  the  main  scale.  Line  i 
on  the  vernier  falls  short  of  a  line  on 
the  limb  by  o.oi  of  an  inch,  line  2 
falls  short  by  0.02  of  an  inch,  and  so  on.  If  the  vernier 


FIG.  9. 


*  Invented  about  1631  by  Pierre  Vernier  of  Burgundy. 


64 


PRINCIPLES.  65 


be  moved  slowly  forward,  the  successive  coincidences 
of  a  line  on  the  vernier  with  one  on  the  limb  will  indi- 
cate successive  advances  of  the  vernier,  each  equal  to 
o.oi  of  an  inch.  If,  then,  the  lines  of  the  vernier  are 
numbered  as  in  Fig.  9,  the  number  on  the  vernier  of  the 
line  which  coincides  will  indicate  the  distance  that  the 
zero  of  the  vernier  has  passed  a  division  of  the  limb. 

65.  Direct  and  Retrograde   Verniers.      In    the   above 
illustration  the  spaces  on  the  vernier  were  shorter  than 
those  on    the    limb,  the    supposed    motion  was    in    the 
direction  of  the  graduation  of  the  limb,  the  successive 
lines    of    the    vernier    came    into    coincidence    in    the 
direction  of  the  graduation,  and  the  numbering  on  the 
limb,  and  also  on  the  vernier,  increased  in    the   same 
direction.       A    vernier    fulfilling    these    conditions    is 
called  a  direct  vernier. 

If  the  spaces  on  the  vernier  are  larger  than  those  on 
the  limb,  and  the  vernier  is  moved  in  the  direction  of 
the  graduation  of  the  limb,  the  successive  coincidence 
will  occur  in  the  direction  opposite  to  the  motion,  and 
also  opposite  to  the  direction  of  the  graduation  on  the 
limb;  and  therefore  the  lines  on  the  vernier  should  be 
numbered  in  an  opposite  direction  to  those  on  the  limb. 
A  vernier  fulfilling  these  conditions  is  called  a  retrograde 
vernier.  For  an  example,  see  Fig.  n,  page  67. 

66.  Least  Count.     The  least  count  of  a  vernier  is  the 
difference  in  length  between  a  space  on  the  limb  and 
one  on  the  vernier.    To  find  the  least  count  of  a  vernier, 
z.e.,  Jo  determine  how  small  a  distance  it  can  measure, 
let  7=  the   length   of   a   division    on   the   limb,  v  —  a 
division   on   the  vernier,  and  n  =  the  total  number  of 
spaces  on  the  vernier.     Then  by  the  principle  of  the 

vernier,  n  I  =  n  v  ±  /,  solving  which  gives  /  —  v  =.  —  = 

the  least  count.     For  example,  in  Fig.  9,  /=  o.i,  and 

/       o.i 

n  =  10:  hence  the  least  count  equals  —  =  --  =  o.oi. 

n        10 


66  VERNIERS.  [CHAP,  v 

The  preceding  formula  expresses  a  very  important 
relation,  since  it  is  the  key  to  reading  all  verniers, 
whether  direct  or  retrograde.  Notice  that  the  lea^t 
count  of  the  vernier  is  equal  to  the  smallest  division 
on  the  limb  divided  by  the  number  of  spaces  on  tne 
vernier.  In  practice,  it  is  not  necessary  to  count  n  • 
it  is  indicated  by  the  numbering  on  the  vernier  itse«r. 
For  exarnple,  if  the  limb  is  divided  to  half-degrees  aua 
there  are  thirty  spaces  on  the  vernier  (which  would  be 
indicated  by  the  end  line  being  numbered  thirty),  the 
vernier  reads  to  one  thirtieth  of  a  half-degree  or  to 
minutes. 

67.  To  READ  A  VERNIEE.      Look  at  the  zero  line  of 
the  vernier.     If  it  coincides  with  a  division  of  the  limb, 
the  number  of  that  line  on  the  limb  is  the  correct  read- 
ing, the  vernier  divisions  not  being  required;  but  if,  as 
usually  happens,  the  zero  of  the  vernier  comes  between 
two  divisions  of  the  scale,  note  the  nearest  division  on 
the  limb  next  less,  and  then  look  along  the  vernier  till 
a    line  is  found   which   exactly  coincides  in   direction 
with  some  line  on    the  limb.     The  number  of  this  line 
on  the  vernier  is  the  distance  between   the  zero  of  the 
vernier  and  the  next  lower  division  of  the   limb,  and 
must   be  added   to  the  reading  taken  from   the  limb. 
The  particular  division  on  the  limb  that  may  be  in  coin- 
cidence is  of  no  consequence. 

68.  Examples.     A   number  of  examples  will  now  be 
given    to    further   illustrate    these   general    principles. 
The  student  should  draw  the  limb  and  the  scale  on  sep- 
arate slips  of  paper  or  card-board,  and  move  one  beside 
the  other  until  he  can  read  them  in  any  position. 

The  vernier  shown  in  Fig.  10  is  the  one  used  on  the 
New  York  leveling  rod.  The  main  scale  is  divided  to 
feet,  tenths,  and  hundredths,  and  ten  divisions  of  the 
vernier  are  equal  to  nine  of  the  limb  ;  therefore  the 
vernier  reads  to  tenths  of  a  hundredth,  or  to  thou- 


TO    READ    A    VERNIER. 


sandths  of  a  foot.  The  vernier  as  drawn  reads  3  feet, 
o  tenths,  3  hundredths,  and  6  thousandths,  or  3.036 
feet. 

Fig.  ii   is  a  retrograde  vernier  which  reads  to   hun- 
aredths  of  an  inch,  the  limb  being  divided  to  inches 


FIG.  10. 


FIG.  ii. 


and  tenths.  It  is  sometimes  applied  to  barometers. 
The  reading  is  15.64  inches. 

Fig.  12  shows  part  of  a  circle  graduated  to  degrees 
and  half-degrees.  The  vernier  has  thirty  parts,  and 
therefore  it  reads  to  minutes.  The  reading  is  212°  30' 
-f-  ii'  —  212°  41'.  This  is  a  very  common  vernier  for 
engineering  instruments. 

The  graduation  of  transits  usually  has  two  rows  of 
numbers,  one  increasing  to  the  right  and  the  other  to 
the  left,  in  which  case  there  are  two  verniers — one  for 
each  series  of  numbers.  Such  an  arrangement  is  shown 
in  Fig.  13,  and  is  called  a  double  vernier.  The  reading 
of  the  left-hand  vernier  and  the  inside  row  of  numbers 
is  182°  40',  while  that  of  the  right-hand  vernier  and  the 
outer  row  of  numbers  is  177°  20'. 


68 


VERNIERS. 


[CHAP,  v 


With  a  double  vernier  there  is  sometimes  a  doubt 
as  to  which  vernier  should  be  read.  To  settle  this, 
notice  whether  the  vernier  divisions  are  larger  or 
smaller  than  those  on  the  limb.  If  the  spaces  on  the 


FIG.  12. 


vernier  are  the  smaller,  it  is  a  direct  vernier,  and  that 
vernier  should  be  read  the  numbers  of  which  increase 
in  the  same  direction  as  those  on  the  limb.  If  the 
spaces  on  the  vernier  are  the  larger,  it  is  a  retrograde 


FIG.  13. 

vernier,  and  that  vernier  should  be  read  the  numbers 
of  which  increase  in  the  opposite  direction  to  those  on 
the  limb.  Or  the  vernier  to  be  read  can  always  be  de- 
termined as  follows:  Move  the  vernier,  until,  say,  the 
ten  line  of  one  vernier  and  the  twenty  of  the  other 
each  coincide  with  a  line  on  the  limb;  look  at  the  zero, 
estimate  the  reading,  and  then  read  the  vernier  that 
agrees  most  nearly  with  the  estimated  reading.  In 
Fig.  13  the  numbers  on  the  vernier  are  inclined  in  the 
same  direction  as  the  numbers  on  the  limb  to  which  the 


TO    READ    A    VERNIER. 


69 


vernier  belongs.  Instruments  are  not  generally  made 
in  this  way,  but  such  an  innovation  would  be  a  great 
improvement. 


FIG.  14. 

Fig.    14   is   another   form   of   double   vernier.     It   is 
called  a  double  folded-vernier,  and  is  often   applied  to 
the  compass,  to  be  used  in   setting  off  the  declination. 
Notice  that  this  form  is  only  half 
as   long   as   the  double  vernier  of 
Fig.  13.     It  is  used  where   there  is 
not  space  enough    for   the    longer 
The  lower  numbers  on   one 


one. 

side  of  the  zero  and  the  upper  ones 
on  the  other  side  constitute  a 
vernier.  The  proper  vernier  to  be 
read  in  any  given  case,  can  be 
determined  by  either  of  the  rules 
of  the  preceding  paragraph.  The 
vernier  as  drawn  reads  2°  2'  to  the 
right.  Notice  that  the  reading  of 
the  vernier  is  2',  and  not  28'. 

Fig.  15  represents  the  scale  and 
vernier  of  the  mountain  barometer. 
The  limb  is  .divided  into  inches, 
tenths,  and  half-tenths  or  five- 
hundredths.  There  are  25  spaces 
on  the  vernier,  and  therefore  it 
reads  to  (.05  —  25  =  .002)  two  thousandths  of  an  inch. 
The  reading  is  28.75  +  0.020  =  28.770  inches.  Notice 


FIG.  15. 


VERNIERS. 


[CHAP,  v 


the  method  of  numbering  the  vernier.  The  numbers 
on  it  indicate  hundredths.  This  form  is  much  more 
convenient  to  read  than  if  the  numbers  ran  from  o 
to  25. 

Fig.  16  shows  the  usual  scale  and  vernier  of  the  bet- 
ter sextants.  The  limb  is  graduated  to  10  minutes,  the 
vernier  has  60  spaces,  and  therefore  it  reads  to  sixtieths 
of  10  minutes,  or  to  10  seconds.  The  numbers  on  the 


FIG.  16. 

vernier  indicate  minutes.  The  reading  is  70°  oo'  on  the 
limb,  plus  5'  10"  on  the  vernier,  or  70°  5'  10". 

69.  PRACTICAL  HINTS.  The  most  frequent  error  in 
reading  a  vernier  is  to  omit  part  of  the  reading  of  the 
limb  ;  as,  for  example,  in  Fig.  12,  forgetting  to  record 
the  half-degree  from  the  limb.  With  circular  arcs  hav- 
ing two  rows  of  figures  there  is  great  danger  of  read- 
ing the  wrong  row. 

To  determine  whether  a  particular  line  on  the  vernier 
coincides  exactly  with  one  on  the  limb,  notice  the  next 
line  on  either  side,  and  see  whether  both  fall  short 
(or  overreach)  equal  amounts.  When  several  lines  of 
the  vernier  appear  to  coincide  equally  with  several 
lines  of  the  limb,  take  the  middle  line.  When  none  of 
the  lines  exactly  coincide,  but  one  line  on  the  vernier  is 
on  one  side  of  a  line  on  the  limb  and  the  next  line  on 
the  vernier  is  as  far  on  the  other  side  of  a  line  on  the 
limb,  the  true  reading  is  midway  between  the  readings 
indicated  by  these  two  lines.  If  the  graduation  is  very 


PRACTICAL    HINTS.  71 


accurate  and  the  lines  fine,  it  is  possible  by  this  method 
to  estimate  the  reading  to  a  half,  or  even  to  a  third,  of 
the  least  count  (§  66). 

It  frequently  happens  that  the  instrument  is  to  be 
used  to  lay  off  a  number  of  equal  angles,  as,  for  ex- 
ample, i°.  The  vernier  is  then  to  be  set  each  time  at 
some  particular  line.  If  it  is  a  double  vernier,  set  the 
zero  line  to  coincide  each  time,  and  notice  whether  the 
next  lines  on  either  side  fall  short  equal  amounts  ;  if 
it  is  a  single  vernier,  set,  say,  the  fifteen  line  to  coincide, 
and  note  the  agreement  on  both  sides.  This  method  is 
considerably  more  accurate  than  reading  the  vernier  by 
looking  at  only  one  line. 

70.  Transits  are  sometimes  made  with  two  verniers, 
one  of  which   reads  to  decimals  of  a  degree   and   the 
other  to  minutes,  as  usual.     A  decimal  vernier  is  very 
convenient  in  laying  out  railroad  curves,  besides  having 
some  minor  advantages  in  other  kinds  of  work. 

71.  MICROMETER.     When  the  highest  degree  of  accu- 
racy is  desired,  a  micrometer  is  used.     A  micrometer 
consists  of  a  fine  screw,  having  a  graduated  head,  which 
moves  a  frame  on  which  is  a  fine  line  or  spider's  thread. 
The  movement  of  the  line  or  thread  is  observed  through 
a  microscope.     The  distance  is  determined  by  the  revo- 
lutions and  fractions  of  a  revolution  of  the  screw. 

Micrometers  are  not  used  on  ordinary  engineering 
tield-instruments. 


CHAPTER   VI. 
OPTICAL   PARTS   OF  THE  TELESCOPE. 

ART.  1.     CONSTRUCTION. 

72.  A  TELESCOPE  consists,  optically,  of  certain  lenses 
which  assist  the  eye  in  seeing  distant  objects  ;  and,  me- 
chanically, of  certain  parts  which  facilitate  the  use  of 
the  optical  parts.     The  mechanical  parts  can  best  be 
discussed  in  connection  with  the  instrument  to  which 
the  telescope  is  applied;  and  hence  in  the  present  chap- 
ter only  the  optical  parts  will   be   considered.     No  at- 
tempt at  an  elaborate  discussion  of   the  theory  of  the 
optical  workings  of  the  telescope  will  be  made;  but  at- 
tention will  be  confined  to  such   points  as  need  to  be 
understood  by  the  engineer  for  the  intelligent  use  of 
his  instruments. 

73.  KINDS  OF  TELESCOPE.     There  are  two  forms  of  the 
simple    refracting    telescope, — usually   known    as    the 
Galilean,  and  the  astronomical.     The  last  term  is  not  a 
happy  one,  and  measuring  is  suggested  as  being   more 
appropriate,  as  will  appear  farther  on. 

The  Galilean  telescope  consists  of  a  double  convex 
lens,  called  the  object-glass  or  objective,  placed  next  to 
the  object,  and  a  double  concave  lens,  called  the  eye- 
piece or  ocular,  placed  near  the  eye.  The  chief  pur- 
pose of  the  objective  is  to  increase  the  amount  of  light 

72 


ART.   l]  CONSTRUCTION.  73 

which  reaches  the  eye  from  the  object  viewed.  The 
sole  object  of  the  eye-piece  is  to  magnify  the  thing 
looked  at.  The  Galilean  telescope  assists  the  eye  by 
its  magnifying  and  light-gathering  power ;  but  such 
a  telescope  would  be  useless  for  making  precise  meas- 
urements, since  there  is  no  means  of  indicating  the 
exact  point  at  which  the  telescope  is  sighted.  It  would 
not  be  inappropriate  to  call  it  a  seeing  telescope.  The 
first  telescope  was  of  this  form,  and  took  its  name  from 
the  inventor,  Galileo.  It  shows  objects  erect.  An 
opera-glass  is  a  good  example. 

The  measuring  telescope  consists  of  three  essential 
parts  :  i,  a  convex  objective,  which  collects  the  rays  of 
light  and  forms  a  bright  inverted  image  of  the  object  ; 
2,  a  convex  eye-piece,  which  is  essentially  a  microscope, 
for  viewing  the  image  formed  by  the  objective  ;  and  3, 
two  fine  wires  or  spider  threads,  placed  in  the  plane  of 
the  image,  the  intersection  of  which  indicates  the  pre- 
cise point  sighted  at.  The  objective  collects  the  light, 
the  eye-piece  magnifies,  and  the  cross  hairs  indicate  the 
point  at  which  the  telescope  is  directed.  If  such  a  tele- 
scope neither  magnified  nor  increased  the  illumination, 
it  would  still  be  of  great  advantage  in  making  measure- 
ments. This  form  shows  the  object  inverted.  Nearly 
all  telescopes  belong  to  this  class. 

Both  of  the  above  skeleton  forms  of  telescope  are 
very  imperfect.  The  methods  of  improving  the  optical 
qualities  of  these  elementary  forms  will  now  be  con- 
sidered. 

74.  THE  OBJECTIVE.  A  single  lens  used  as  an  objec- 
tive has  the  following  defects:  i.  The  rays  of  light 
which  traverse  a  spherical  lens  near  the  edge,  are  re- 
fracted to  a  point  nearer  the  lens  than  the  rays  which 
pass  through  the  central  portion  ;  consequently  the 
image  is  blurred.  This  deviation  of  the  rays  from  the 
focus  is  called  spherical  aberration.  2.  Rays  of  white 


74  OPTICAL   PARTS   OF    THE    TELESCOPE.         [CHAP.  VI 

light,  in  passing  a  single  lens,  are  resolved  into  the 
colors  of  the  prismatic  spectrum  ;  consequently  the 
image  will  be  disfigured  by  colored  light.  This  devi- 
ation of  the  different  colored  rays  is  called  chromatic 
aberration. 

The  objective  of  a  telescope  is  rendered  almost  free 
from  these  defects  by  substituting  for  the  single  Jens,  a 
compound  one  composed  of  a  double-convex  crown- 
glass,  and  a  concavo-convex  flint-glass,  lens.  The  two 
components  have  different  refractive  and  dispersive 
powers,  and  by  giving  the  four  surfaces  proper  curva- 
tures the  above  defects  may  be  nearly  eliminated. 

75.  THE  EYE-PIECE.     A  single   lens   used   as   an   eye- 
piece possesses  the  same  defects — spherical  and  chro- 
matic aberration — as  when  used  as  an  objective,  but  in 
a  less  degree.     A  single  lens  as  an  eye-piece  has  also  the 
following  defects  :   i.  The  image  of  a  flat  object  formed 
by  a  lens  does  not  lie  in  a  plane,  but  is  concave  towards 
the  lens.     This  deviation  of  the  image  from  a  plane  is 
termed  aberration  of  sphericity,  which  is  wholly  separate 
and    distinct    from    spherical    aberrations  (§  74).     The 
objective  also  possesses  this  defect,  but  in  so  slight  a 
degree  as  to  be  inappreciable  ;  while  in  an  eye-piece  of  a 
single  lens  it  is  very  serious.     2.  A  telescope  with  a  sin- 
gle lens  for  an  eye-piece  has  a  very  limited  field  of  view. 

In  both  forms  of  telescope,  the  chromatic  and  spheri- 
cal aberration,  and  also  the  aberration  of  sphericity  of 
the  eye-piece,  are  nearly  eliminated  by  substituting  two 
plano-convex  lenses  for  the  single  lens,  which  also  in- 
creases the  field  of  view. 

76.  Huyghen  or  Negative  Eye-piece.     This  is  a  modifi- 
cation  of   the  concave  eye-piece  of  the   Galilean  tele- 
scope.     It    consists   of    two  plano-convex  lenses    with 
their  convex  sides  turned  towards  the  objective.     The 
relations  of  the  eye  lenses  to  each  other,  and  to  the  ob- 
jective, are  shown  in  Fig.  17. 


ART.   l] 


CONSTRUCTION. 


75 


P  is  the  object;  O  is  the  objective  ;  and  a  and  b  con- 
stitute the  eye-piece,  a  being  known  as  the  field  lens 
and  b  as  the  eye  lens.  Notice  that  the  eye-piece  is 


FlG.    17. — HUYGHEN   OR    NEGATIVE    EYE-PIECE. 

placed  between  the  objective  and  its  principal  focus, 
F.  The  large  arrow  at  /  represents  the  object  as  it 
appears  through  the  telescope  ;  the  small  one  repre- 
sents it  as  it  would  appear  without  the  instrument. 
The  angle  which  the  large  arrow  at  /  subtends  at  the 
eye,  divided  by  the  angle  which  the  object  subtends  at 
the  eye,  is  equal  to  the  magnifying  power  ;  that  is,  the 
magnifying  power  is  equal  to  the  ratio  of  the  larger 
arrow  at  /to  the  small  one. 

77.  Ramsden  or  Positive  Eye-piece.  This  is  a  modifi- 
cation of  the  convex  eye-piece  of  the  measuring  tele- 
scope. It  consists  of  two  plano-convex  lenses  with 
their  convex  sides  towards  each  other.  The  relations 
of  the  eye  lenses  to  the  objective  are  shown  in  Fig.  18. 


FIG.  18. — RAMSDEN  OR  POSITIVE  EYE-PIECE. 


The  nomenclature  is  as  before.  Notice  that  the  eye 
lenses  are  farther  from  the  objective  than  its  principal 
focus.  The  objective  forms  an  image  at  F  which  is 
viewed  with  the  eye-piece.  The  magnifying  power  is 
the  ratio  of  the  two  arrows  at  /,  as  before. 


76  OPTICAL    PARTS   OF    THE    TELESCOPE.        [CHAP.  VI 

Notice  that  the  essential  difference  between  the  posi- 
tive and  negative  eye-pieces  is  that  only  with  the  former 
is  a  real  image  of  the  object  formed,  and  hence  it  is 
the  only  eye-piece  with  which  cross  hairs  can  be  used. 
Spider  lines  are  sometimes  placed  in  negative  eye- 
pieces, for  example  in  sextants,  simply  to  indicate  the 
middle  of  the  field  of  view  ;  but  they  are  only  indirectly 
concerned  in  the  accuracy  of  the  observations.  The 
negative  eye-piece  is  better  for  simply  seeing,  while  the 
positive  is  absolutely  necessary  in  making  precise 
measurements.  The  positive  eye-piece  is  always  used 
in  transits,  levels,  etc. 

Modifications  of  Ramsden's  eye-piece,  consisting  in 
the  substitution  of  compound  lenses  instead  of  the 
single  lenses  shown  in  Fig.  18,  are  frequently  employed. 
The  principal  ones  are  Kellner's,  which  consists  of  an 
achromatic  combination  instead  of  lens  a  of  Fig.  18, 
and  Steinheil's,  which  consists  of  two  achromatic  com- 
binations instead  of  the  single  lenses  a  and  b  oi  Fig.  18. 
These  eye-pieces  are  better  than  Ramsden's  in  that  they 
are  free  from  color  and  have  a  perfectly  flat  field.  Of 
the  two,  Kellner's  permits  the  larger  field. 

78.  Erecting  Eye-piece.  The  erecting  or  terrestrial 
eye-piece,  in  its  most  elementary  form,  consists  of  a 

<i ;  i       i!  11 

FIG.  19.— ERECTING  EYE-PIECE. 

convex  lens  placed  between  the  eye  and  eye-piece  of 
the  measuring  telescope,  which  inverts  the  image 
formed  by  the  objective  and  shows  the  object  erect ; 
but  in  its  common  form  it  consists  of  a  pair  of  plano- 
convex lenses  instead  of  the  single  lens.  Fig.  19  shows 
the  usual  arrangement  of  the  lenses  and  diaphragms  of 
an  erecting  eye-piece.  The  pair  of  lenses  next  to  the 


ART.   l]  CONSTRUCTION.  77 

eye  is  called  the  erecting  piece  ;  the  pair  next  to  the 
objective  is  the  ordinary  positive  eye-piece.  For  a 
Kellner's  erecting  eye-piece,  see  Fig.  27  (page  100)  or 
Fig.  62  (page  222). 

The  erecting  eye-piece  is  inferior  to  the  inverting 
eye-piece,  owing  to  the  loss  of  light  occasioned  by  the 
two  extra  lenses.  The  inconvenience  of  the  inversion 
of  the  object  is  easily  overcome  with  a  little  practice. 
Furthermore,  other  things  being  equal,  the  telescope 
is  shorter  with  the  inverting  eye-piece — which  is  quite 
an  important  advantage  in  the  transit.  Most  Amer- 
ican engineering  instruments  have  an  erecting  eye- 
piece ;  but  it  would  be  a  great  improvement  if  all 
were  provided  with  the  inverting  eye-piece.  The 
latter  is  much  more  common  in  Europe  than  in 
America. 

79.  TELESCOPE  SLIDES.  To  assist  in  focusing  the  ob- 
jective for  different  distances,  the  object-glass  is 
fastened  in  a  tube  which  slides,  with  a  rack  and  pinion, 
into  the  end  of  the  main  tube  of  the  telescope.  In  in- 
struments provided  with  an  inverting  eye-piece  the 
objective  is  fixed,  and  the  cross  hairs  and  eye-piece 
move  together,  to  and  from  the  objective.  It  is  im- 
material which  form  is  used  ;  the  principle  is  the  same 
in  both,  it  being  only  important  that  the  slide  shall  be 
straight  and  fit  snugly. 

To  facilitate  the  focusing  of  the  eye-piece  upon  the 
cross  hairs,  the  ocular  is  provided  with  a  similar  slide. 
In  some  instruments  the  ocular  is  moved  by  a  rack  and 
pinion  ;  but  this  is  unnecessary  and  even  worse  than 
useless,  for,  having  once  adjusted  the  ocular  for  distinct 
vision  of  the  cross  hairs,  it  needs  no  change  except  for 
different  observers,  and  it  is  better  that  it  should  not 
be  easily  moved.  If  no  rack  and  pinion  is  provided,  the 
ocular  is  moved  in  and  out  by  hand,  with  a  screw-like 
motion. 


FIG.  20. 


\  OPTICAL    PARTS    OF    THE    TELESCOPE.         [CHAP.   VI 

80.   CROSS    HAIRS.*     These    are    usually    two    spider 

threads,  one  vertical  and  the 
other  horizontal,  fastened  to 
a  ring  which  is  adjusted  in 
the  tube  by  four  screws. 
Fig.  20  shows  a  front  view  of 
the  ring  in  place.  BB,  Figs. 
27  (page  100)  and  62  (page 
222)  show  side  views  of  the 
ring.  Lines  ruled  or  etched 
upon  a  piece  of  thin  glass 
are  sometimes  used.  The 
cross  hairs  are  sometimes  made  of  very  fine  platinum 
wire.  Platinum  can  be  drawn  to  the  required  fine- 
ness only  by  being  previously  surrounded  by  silver, 
which  is  removed  by  acid  after  the  wire  is  drawn. 
Both  spider  threads  and  platinum  wires  have  their  ad- 
vantages and  disadvantages,  and  as  a  consequence  their 
advocates  and  opponents. 

For  the  platinum  wires,  it  is  claimed  that  they  are 
best  because  they  are  opaque  ;  and  this  is  a  desirable 
property  when  the  cross  hairs  must  be  illuminated,  as 
in  astronomical  work  and  mine  surveying.  Spider 
lines,  particularly  dark-colored  ones,  can  be  illuminated 
fairly  well.  It  is  claimed  also  that  wires  are  unaffected 
by  the  humidity  of  the  atmosphere,  and  hence  the  line 
of  collimation  is  not  liable  to  change  from  this  cause — 
an  advantage  which  does  not  exist  if  the  spider  threads 
are  properly  stretched.  On  the  other  hand,  it  is 
claimed  that  platinum  wires  lose  their  elasticity  and 
sag,  and  also  that  they  oxidize  or  corrode.  Spider 
threads, on  account  of  their  fineness  and  cheapness. and 
the  facility  with  which  they  may  be  applied,  will  con- 


*  Cross  hairs  were  first  used  by  Picard,  in  1669. 


ART,    l]  CONSTRUCTION.  79 

tinue  in  the  future,  as  in  the  past,  to  be  used  almost 
universally  for  cross  hairs  in  engineering  instruments. 

81.  Stretching  the  Spider  Webs.  It  does  not  require 
much  time  or  skill  to  replace  spider  webs,  the  belief  of 
many  to  the  contrary  notwithstanding.  The  ring 
which  carries  the  cross  hairs  can  be  taken  out  (with 
some  telescopes  only  after  having  taken  out  the  eye- 
piece) by  removing  two  opposite  screws  and  inserting  a 
soft  wooden  stick  of  suitable  size  into  one  of  the  holes 
thus  left  open  in  the  ring  (which  last  is  turned  sideways 
for  that  purpose),  and  then  removing  the  other  screws. 
Two  scratches,  at  right  angles  to  each  other,  will  be 
found  upon  the  face  of  the  ring,  into  which  the  hairs 
are  to  be  fastened. 

The  best  spider  lines  are  those  of  which  the  spider 
makes  its  nest.  These  nests  are  yellowish-brown  balls 
which  may  be  found  hanging  on  the  shrubs,  etc.,  in  the 
late  fall  or  early  winter.  When  found,  the  nest  should 
be  torn  open  and  the  eggs  thrown  out  ;  if  this  is  not 
done,  the  young  spiders  when  hatched  will  eat  the 
threads.  The  fibers  next  to  the  eggs  are  to  be  preferred 
on  account  of  their  darker  color. 

Draw  out  a  single  fiber  and  attach  each  end  of  it  to 
as  heavy  a  weight  as  experiment  shows  it  will  sup- 
port. Dampen  the  thread  by  breathing  upon  it,  by 
holding  it  in  a  current  of  steam,  or,  better,  by  dipping 
it  in  clean  hot  water.  Support  the  ring  in  such  a  way 
that  when  the  thread  is  laid  across  it,  the  weights  may 
hang  freely  down  the  sides,  thus  stretching  the  thread. 
The  thread  may  be  moved  easily  with  a  pin,  and  when 
in  the  proper  position  it  can  be  fastened  with  wax,  gum, 
varnish,  or  the  like,  shellac  varnish  being  the  best  for 
this  purpose.  The  main  point  is  to  stretch  the  web 
thoroughly  and  fasten  it  tightly.  If  the  work  is  well 
done,  the  threads  will  remain  straight  when  the  reticule 
is  placed  in  a  current  of  steam  or  even  in  hot  water. 


8o  OPTICAL    PARTS    OF    THE    TELESCOPE.         [CHAP.   VI 


ART.  2.  TESTING  THE  TELESCOPE. 

82.  In  buying  an  instrument  it  is  very  desirable  to 
test  the  optical  qualities  of  the  telescope  ;  and  the  fol- 
lowing directions  are  sufficient  for  such  examination. 

83.  CHROMATIC  ABERRATION.    To  test  for  chromatic 
aberration    (i,  §  74),    focus   the    telescope   upon    some 
bright  object,  either  a  celestial  body  or  a  white  disk, 
and  then  move  the  object-glass  slowly  in  and  out.     If 
in  the  first  instance  a  light  yellow  ring  is  seen  at  the 
edge  of  the  object,  and  in  the  second  a  ring  of  purple 
light,   the  object-glass   may  be  considered   perfect,   as 
this  proves  that  the  most  intense  colors  of  the  prismatic 
spectrum  (orange  and  blue)  are  corrected. 

84.  SPHERICAL   ABERRATION.     To  test  for  spherical 
aberration  (2,  §  74),  cover  the  object-glass  with  a  ring 
of  black  paper,  so  as  to  reduce  the  area  of  the  aperture 
about   one   half,  and   focus    on    some  small  and    well- 
defined   object   for   distinct  vision.     Then   remove  the 
ring  of  the  paper  and  cover  that  part  of  the  objective 
previously  left  open,  and  notice  how  much  the  object- 
glass   must   be   moved   in    or    out    for    distinct    vision. 
The  amount  of  shift  measures    the  spherical    aberra- 
tion of  the  objective.     Very  little,  if  any,  motion  should 
be  required  to  obtain  a  distinct  view.     Another  test  is 
to  focus   sharply   upon    some   object,    when    the   least 
motion  of  the  objective  should  render  the  object  indis- 
tinct.    This  last  is  not  as  good  a  test  as  the  former,  for 
the  eye  will  change  focus  slightly  to  accommodate  itself 
to  the  change  of  distance. 

85.  DEFINITION.      The  definition   of   a  telescope  de- 
pends upon  the  accuracy  of  the  curvature  of  the  sur- 
faces of  the  several  lenses,  and  upon   the  centering,  i.e., 
the  coincidence  of  the  axes  of  the  component  lenses  of 


ART.    2]  TESTING    THE    TELESCOPE.  8l 

the  objective  and  ocular.  The  reader  should  distinguish 
between  illumination  and  definition.  The  lack  of  the 
former  causes  the  image  to  be  faint  ;  the  lack  of  the 
latter  causes  the  image  to  be  indefinite.  Indistinctness 
may  be  caused  by  spherical  aberration  ;  and  therefore 
the  test  for  that  should  precede  the  test  for  definition. 

To  test  a  telescope  for  definition,  focus  upon  some 
small,  well-defined  object — small,  clear  print  is  the  best. 
.If  the  print  appears  clear  and  well  defined  and  fully  as 
legible  at  40  or  50  feet  as  if  viewed  with  the  naked  eye 
at  8  or  10  inches  (the  best  distance  for  distinct  vision), 
the  surfaces  of  the  lenses  are  correct  and  well  finished. 

To  determine  whether  the  lenses  are  well  centered, 
fix  a  white-paper  disk  having  a  diameter  of,  say,  an 
eighth  of  an  inch  and  a  sharp  outline,  in  the  middle  of  a 
piece  of  black  paper  or  cloth.  Place  this  in  a  good 
light  30  or  40  feet  from  the  telescope;  then  if  the 
image  of  the  disk,  when  a  little  out  of  focus,  is  equally 
surrounded  on  all  sides  by  a  uniform  haze,  the  center- 
ing is  good. 

Other  things  being  the  same,  the  lower  the  magnify- 
ing power  the  better  the  definition. 

86.  FLATNESS  OF  FIELD.  The  flatness  of  the  field 
depends  mainly  upon  the  aberration  of  sphericity  of 
the  eye-piece  (§  75).  To  test  a  telescope  in  this 
respect,  draw  with  India  ink  a  heavy-lined  square,  6 
or  8  inches  on  a  side,  on  a  sheet  of  white  paper,  and 
fasten  the  paper  to  a  flat  board.  Place  this  object  at 
such  a  distance  that  the  square  shall  nearly  fill  the 
field.  Focus  the  telescope  on  it ;  then,  if  the  sides 
appear  perfectly  straight,  the  telescope  is  perfect  in 
respect  to  flatness  of  field. 

A  telescope  which  distorts  the  image  perceptibly 
causes  no  error  in  common  use,  but  is  decidedly 
objectionable  for  stadia  measurements,  where  two 
points  of  the  field  are  used  at  the  same  time. 


82  OPTICAL    PARTS    OF    THE    TELESCOPE.        [CHAP.    VI 

87.  SIZE  OF  THE  FIELD.  By  the  field  of  view  is  meant 
all  those  points  which  are  visible  at  the  same  time 
through  the  telescope.  In  Fig.  21  O  is  the  objective, 
E  the  ocular,  DABC  the  object,  ab  the  image  of  AB. 
The  two  extreme  rays  from  A  and  B  just  strike  the 
edge  of  the  ocular,  and  rays  from  C  and  D  do  not 
enter  the  ocular  ;  therefore  the  angle  AOB  is  the  field 


FIG.  21. 

of  view.  Notice  that  it  is  independent  of  the  size 
of  the  object-glass,  and  varies  (i)  inversely  as  the 
distance  between  the  objective  and  the  ocular,  and 
(2)  directly  as  the  size  of  the  eye-piece.  The  greater 
the  magnifying  power  the  less  the  field  of  view.  The 
wider  the  field  of  view  the  better,  since  width  of  field 
facilitates  rapid  working.  One  of  the  advantages  an 
eye-piece  composed  of  two  plano-convex  lenses  has 
over  a  single  lens  is  the  larger  field  of  view. 

To  determine  the  angular  width  of  field,  sight  upon 
any  object,  and  mark  the  two  opposite  sides  of  the 
field.  The  distance  between  these  points  multiplied 
by  57.3  and  divided  by  the  distance  from  the  object  to 
the  objective,  is  the  angular  width  of  field,  in  degrees. 
Since  the  field  varies  with  the  distance  (particularly  if 
it  be  short),  a  considerable  distance,  say  200  or  300 
feet,  should  be  employed  in  making  this  test. 

The  field  of  view  varies  with  the  maker,  but  for  the 
telescopes  on  ordinary  engineering  instruments  it  is 
about  as  follows  :  for  a  magnifying  power  of  20,  i°  30'; 

25,  i°  15';  30,  i°;  35>  5°'- 

88.  APERTURE  OF  OBJECTIVE.  By  the  aperture  of  the 
objective  is  meant  the  effective  diameter  of  the  object- 
glass,  t,e.t  that  part  of  the  objective  which  transmits 


ART.    2]  TESTING    THE    TELESCOPE.  83 

light  that  finally  reaches  the  eye.  Usually,  diaphragms 
are  placed  in  the  tube  of  the  telescope  with  a  view  to 
improve  the  optical  qualities  of  the  lenses.  If  the 
diaphragm  is  placed  near  the  object-glass,  it  will  cut 
off  those  rays  which  pass  through  the  objective  near 
the  edge,  thus  improving  the  definition — without 
diminishing  the  field  but  with  a  loss  of  illumination  ; — 
but  if  the  diaphragm  is  placed  in  or  near  the  eye-piece, 
it  may  diminish  both  the  illumination  and  the  field  of 
view. 

To  find  the  real  aperture,  direct  the  telescope  towards 
the  sky,  and  place  a  pointer  in  contact  with  the  ob- 
jective so  that  the  pointer  may  be  seen  in  the  small 
illuminated  circle  which  will  be  noticed  at  the  small 
opening  of  the  eye  end  when  the  head  is  drawn  back 
a  short  distance  from  the  telescopy.  Then  move  the 
pointer  over  the  face  of  the  object-glass  until  the  point 
just  disappears,  and  measure  the  distance  from  the 
pointer  to  the  edge  of  the  object-glass.  The  real  or 
clear  aperture  of  the  objective  is  equal  to  the  diameter 
of  the  object-glass  minus  twice  the  above  distance. 
In  making  this  observation  a  hand  magnifier  is  of  great 
assistance  in  viewing  the  image  of  the  objective. 

A  better  method  of  finding  the  real  aperture  is  as 
follows:  Fix,  with  water  or  thin  gum,  several  small 
pieces  of  opaque  paper,  say  0.05,  o.io,  and  0.15  of  an 
inch  square,  in  a  row  around  the  very  outer  edge  of 
the  objective.  Then  turn  the  instrument  up  to  the  sky, 
and  observe  as  before.  If  all  the  pieces  can  be  seen, 
then  the  aperture  is  equal  to  the  entire  objective  ;  but 
if  any  piece  is  invisible,  the  margin  of  the  glass  is  cut 
off  to  this  extent. 

As  it  is  more  difficult  to  get  the  outer  portion  of 
the  objective  to  correct  curvature,  it  is  not  uncom- 
mon to  cut  off  this  portion  with  a  diaphragm,  which 
renders  the  objective  equal  only  to  a  smaller  glass. 


84  OPTICAL   PARTS   OF    THE   TELESCOPE.        [CHAP.   VI 

Even  an  apparently  small  margin  cut  off  makes  a 
considerable  difference  in  the  light-collecting  power. 
For  example,  a  i^-inch  objective  will  collect  56  per 
cent  more  light  than  a  i-inch  glass.  Not  infrequently 
objectives  are  reduced  in  as  great  a  proportion  as  in 
this  example. 

The  aperture  of  the  telescopes  on  ordinary  engineer's 
transits  is  i  to  if  inches,  and  on  levels  ij  to  i£  inches. 

89.  MAGNIFICATION.     The    power   of    a    telescope,    or 
degree    of    magnification,    depends    upon    the    relative 
focal  lengths   of   the   objective   and    of.  the   eye-piece. 
Mathematically,  any  power  can   be  given  to  any  tele- 
scope ;  but,  in  practice,  it  is  limited  by  the  effects  of 
loss   of   light,   size   of   field,    and    imperfection    of   the 
lenses.     For  rapid  work,  the  exact  focusing  necessary 
with  high  powers  is  a  drawback,  since  a  small  change 
in  distance  requires  a  corresponding  change  in  focus. 
The  magnifying  power  varies  slightly  with  the  distance 
to  the  object  ;   but,  fortunately,  the  exact  magnifying 
power  of  the  telescopes  on  engineering  instruments  is 
not  required.     There  are  several  methods  of  measuring 
the  magnifying  power,  of  which  the  two  following  are 
the  simplest. 

90.  First  Method.     If  the  telescope  be  directed  toward 
the  open  sky  in    the   day-time    and    the    eye    be   held 
8  or  10  inches  back  from  the  eye  end,  a  small  bright 
illumined   circle  will  be   seen,  which  is   nothing   more 
than  the  image  of  the  objective  opening  of  the  tele- 
scope.    Measure    the    diameter    of    this    circle    with    a 
finely  graduated   scale  of  equal   parts.     Also   measure 
the  real  aperture  of  the  objective   (§  88).     Then   the 
magnifying  power  is  equal  to  the  quotient  arising  from 
dividing  the  diameter  of  the  aperture  of  the  object-glass 
by  the  diameter  of  the   illuminated  circle.     The  chief 
difficulty  in  this  method  lies  in  the  exact  measurement 
of  the  diameter  of  the  small  illuminated  circle, 


ART.  2]  TESTING    THE    TELESCOPE.  85 

91.  Second  Method.     Let  a  staff  which  is  very  boldly 
divided    into    equal    parts    by   heavy   lines   be   placed 
vertically  at   any  convenient   distance  from    the   tele- 
scope— for  example,  150  feet.    While  one  eye  is  directed 
towards  the  staff  through  the  telescope,  the  other  may 
observe  the  staff  by  looking  along  the  outside  of  the 
tube.     One  division  of  the  staff  will  be  seen  by  the  eye 
at    the   eye-piece    to    be    magnified    so   as    to   cover   a 
number   of    divisions    of    the   staff,   and    this    number, 
which    is    the    magnifying    power    required,    may    be 
observed  by  looking  with  the  other  eye  along  the  outside 
of  the  tube.     A  little  difficulty  may  be  experienced  on 
the  first  trial  of  this  method,  but  after  a  few  attempts  it 
becomes  very  easy. 

The  magnifying  power  of  the  telescopes  on  ordinary 
engineer's  transits  varies  from  fifteen  to  thirty,  twenty 
to  twenty-five  being  the  more  common  on  good  instru- 
ments ;  and  for  levels  it  varies  from  twenty  to  fifty, 
twenty-five  to  thirty  being  the  more  common.  The 
higher  the  power  the  greater  the  minimum  distance  at 
which  an  object  can  be  seen  through  the  telescope.  It 
is  frequently  desirable,  particularly  with  a  transit,  to 
focus  on  a  point  only  4  to  6  feet  from  the  instrument. 
This  can  usually  be  accomplished  with  a  magnifying 
power  of  twenty. 

92.  ILLUMINATION.     The  brightness  of  an  object  seen 
through  a  telescope  depends  upon  (i)  the  size  of  the 
objective,  (2)  the  polish  and  transparency  of  the  lenses, 
(3)  the  magnifying  power  of  the  telescope,  and  (4)  the 
size  of  the  pupil  of  the  eye. 

1.  Obviously,  the  quantity  of  light  entering  a  tele- 
scope is  dependent  upon  the  size  of  the  aperture  of  the 
object-glass,  for  the  larger  its  area   the   more  rays  of 
light  proceeding  from  any  point  of  the  object  will  be 
intercepted  and  transmitted. 

2.  In  ever    telescope  light  is  lost  by  reflection  from 


86  OPTICAL    PARTS    OF    THE    TELESCOPE.         [CHAP.   VI 

the  surfaces  of  the  lenses,  and  by  imperfect  trans- 
parency. It  is  probable  that  no  engineering  instrument 
transmits  over  85  per  cent*  of  the  light  striking  the 
objective,  and  many  of  them  transmit  much  less. 

3.  The    object-glass     transmits  a  certain   amount   of 
light,  which  the  eye-piece  distributes  over  a  larger  or 
smaller  area,  according  to  the  magnifying  power  of  the 
telescope;  therefore  the  brightness  of  the  image  varies 
inversely  as  the  square  of  the  magnifying  power.    Since 
brightness    of   view    is    an    indispensable    requisite    of 
a  good  telescope,  the  magnification    should    never    be 
excessively  large,  as    it    frequently    happens    that    the 
telescope  is  used  in  viewing  objects  only  faintly  illu- 
mined. 

4.  In  order  that  all  the  light  passing   through    the 
telescope  may  be   received  by  the  eye,  the  beam  that 
emerges  from  the  eye  end  must  not  exceed  the  diameter 
of  the  pupil  of  the  eye;   for  if  the  emergent  beam  is 
larger,    part   of   it  can   not    enter    the   eye,   and    conse- 
quently part  of  the  light  intercepted  by  the  objective 
will   be  lost.     The   average  size  of  the  pupil   is  about 
one  tenth  of  an  inch;    and  hence  the  diameter  of  the 
emergent  beam  should  not  be  more.     The  best  possible 
effect  will  be  when  the  beam   is  of  exactly  the   same 
diameter  as  the  pupil,  for  the  brightness  of  the  object 
will  then  be  the  same  (except  for  the  losses  referred  to 
in  paragraph  2,  above)  as  when  viewed  with  the  naked 
eye.     If  the  emergent  beam   is  smaller  than  this,  the 
brightness   will   be    less    than    when    viewed  with    the 
naked  eye. 

93.  Since  there  can  be  no  unit  by  which  to  measure 
the  illuminating  power  of  a  telescope,  we  can  find  only 
the  relative  illumination.  If  two  telescopes  are  to  be 
compared  as  to  light,  they  should  stand  side  by  side  and 

*  Chauvenet's  Practical  Astronomy,  vol.  ii,  p.  17, 


ART.  2]  TESTING    THE    TELESCOPE.  .  87 

be  looked  through  at  the  same  time,  so  as  to  be  under 
the  same  atmospheric  conditions. 

If  two  telescopes  have  the  same  aperture,  and  also 
the  same  magnifying  power,  they  may  be  compared  by 
placing  them  side  by  side,  and,  as  night  approaches,  ob- 
serving the  same  object  through  each.  The  one  through 
which  the  object  is  longest  visible  has  the  better  illu- 
mination. Of  course,  both  observations  must  be  made 
by  the  same  person.  Instead  of  waiting  for  the  ap- 
proach of  night,  observe  the  distance  at  which  fine 
print  can  be  read  with  each,  or  the  distance  at  which 
the  time  can  be  read  from  the  second  hand  of  a  watch. 
Notice  that  the  last  method  involves  the  definition 
(§  85)  and  spherical  aberration  (§  84)  of  the  telescope, 
as  well  as  the  illumination. 

If  the  two  telescopes  do  not  have  equal  apertures  nor 
equal  magnifying  powers,  a  numerical  expression  for 
the  illuminating  power  may  be  obtained  as  follows:  By 
observation  determine  the  maximum  distance  at  which 
time  can  be  told  from  the  face  of  a  watch,  through  each 
telescope.  Then,  if  the  lenses  are  equally  transparent, 
these  distances  should  be  to  each  other  directly  as  the 
clear  apertures,  and  inversely  as  the  magnifying  pow- 
ers. That  is,  if  d^  and  d^  represent  the  distances,  al  and 
#a  the  apertures,  and  m^  and  m%  the  magnifying  power, 
then 


The  error  of  observation  by  this  method  is  considerable, 
owing  to  the  inability  of  the  eye  to  judge  accurately  of 
equal  illuminations,  and  because  the.  method  also  in- 
volves the  defining  powers  of  the  two  instruments. 


88  OPTICAL   PARTS  OF   THE   TELESCOPE.        [CHAP.  VI 


ART.  3.     USING  THE  TELESCOPE. 

94.  ADJUSTMENT  FOE  PARALLAX.  Parallax  is  an  appar- 
ent movement  of  the  cross  hairs  in  reference  to  the 
object  sighted,  caused  by  a  real  movement  of  the  eye 
of  the  observer.  It  shows  that  the  image  and  cross 
hairs  are  not  in  the  same  plane.  Of  course,  if  the 
object  changes  its  position  for  different  positions  of  the 
eye,  it  will  be  impossible  to  do  accurate  work  with  the 
.telescope.  Therefore  a  telescope  should  be  accurately 
adjusted  for  parallax  before  being  used  in  precise 
measurements.  All  measuring  telescopes  require  this 
adjustment.  It  consists  in  bringing  the  cross  hairs  and 
the  image  exactly  into  the  same  plane.  In  making  this 
adjustment  two  steps  are  required:  (i)  focusing  the 
cross  hairs,  and  (2)  focusing  the  object. 

1.  To  focus  the  cross  hairs,  direct  the  telescope  to- 
ward the  sky,  or  throw  it  out  of  focus  so  that  no  object 
can  be  distinguished  in  the  field,  and  move  the  ocular 
in  or  out  until   the  cross  hairs  can  be  seen  very  dis- 
tinctly.    When    the   cross   hairs  are    properly  focused, 
little  specks  of  dust  may  be  seen  on  them.     No  object 
should  be  in  the  field  of  view  while  this  adjustment  is 
being   made,  for   the   eye   is   continually  changing   its 
focus  to  accommodate  itself  to  the  distance  of  the  object 
viewed.     Before  pronouncing  the   adjustment   correct, 
close   the   eye   for   a   moment,   and    then    sight   again. 
Having  adjusted  the  eye-piece,  it  need  not  be  changed 
except  for  the  change  of  focus  of  the  eye  with  advanc^ 
ing  age,  or  for  different  observers.     The  rack  and  pinion 
frequently    provided    for    focusing    the   cross    hairs    is 
worse  than  useless. 

2.  To  focus  the  objective,  direct  the  telescope  to  the 
object,  keeping  the  attention  fixed  upon  the  cross  hairs 
so  that  the  eye  shall  not  change  focus  to  accommodate 


ART.  3]  USING  THE  TELESCOPE.  89 

itself  to  the  position  of  the  object,  and  move  the  objec- 
tive in  or  out  until  the  object  appears  very  sharply 
defined.  Then  move  the  eye  back  and  forth  sidewise, 
and  note  whether  the  cross  hairs  and  object  alter  their 
relative  position.  If  the  cross  hairs  appear  to  move  with 
the  eye,  they  are  farther  from  the  eye  than  the  image, 
and  therefore  the  objective  should  be  moved  nearer  the 
object  ;  for  remember,  first,  that  the  farther  object  ap- 
pears to  move  with  the  eye,  and,  second,  that  the  farther 
the  object  from  the  objective  the  nearer  is  the  image. 
This  test  is  more  easily  made  than  described.  When 
the  objective  is  properly  focused  there  should  be  abso- 
lutely no  movement  of  the  cross  hairs  with  reference 
to  the  object. 

Many  of  the  errors  of  instrument  work  are  due  to 
parallax  ;  and  this  source  of  error  is  more  serious  as  the 
work  becomes  more  accurate.  The  higher  the  power 
the  more  difficult  it  is  to  eliminate  parallax. 

95.  CAKE  or  THE  TELESCOPE.  If  the  objective  or  the 
lens  next  to  the  eye  becomes  dusty,  brush  it  with  a  fine 
camel-hair  brush,  or  rub  it  with  a  piece  of  soft,  clean 
chamois-skin  or  a  piece  of  old  linen  or  silk,  taking  care 
to  use  a  clean  spot  for  each  rub.  Unnecessary  rubbing 
of  the  lenses  should  be  avoided,  since  it  will  destroy  the 
fine  polish  upon  which  depends  the  sharpness  and 
brilliancy  of  the  image.  Dust  upon  the  glasses  is  not 
as  objectionable  as  a  thin,  or  even  almost  imperceptible, 
film  of  grease  ;  therefore  the  lenses  should  never  be 
touched  with  the  fingers.  When  the  lenses  become 
very  dirty,  wash  them  with  alcohol. 

The  interior  face  of  the  objective  and  the  interior 
lenses  of  the  eye-piece  will  seldom  need  cleaning,  unless 
water  should  find  access  to  the  inside  of  the  tube.  Having 
removed  the  cell  containing  the  objective,  care  should 
be  taken  to  screw  it  back  exactly  to  its  former  position, 
else  the  adjustment  of  the  line  of  sight  may  be  de- 


90  OPTICAL    PARTS   OF    THE    TELESCOPE.        [CHAP.  VI 

stroyed.  The  component  lenses  of  the  objective  should 
never  be  taken  apart  nor  removed  from  the  cell  contain- 
ing them,  since  they  may  not  be  returned  to  their  for- 
mer position,  thus  disturbing  their  adjustments  ;  but  if 
they  should  accidentally  become  separated,  be  sure  to 
replace  them  with  the  double-convex  crown-glass  out- 
ward. Dust  should  be  carefully  kept  from  the  inside 
of  the  telescope  tube;  if  not,  it  will  get  on  the  lenses 
and  cross  hairs.  When  not  in  use  the  eye-piece  and 
object-glass  should  be  covered  by  their  caps. 

If  the  telescope  slide  should  get  to  fretting  or  cutting, 
take  it  out  and  smooth  the  rough  places.  The  blade  of 
a  penknife  forms  a  very  good  instrument  for  this  pur- 
pose. Scrape  with  the  edge,  slightly  inclining  it,  and 
burnish  with  the  back  of  the  knife.  Wipe  out  the  in- 
side of  the  tube,  and  if  possible  burnish  and  scrape  that 
smooth.  Grease  the  slide  slightly,  and  wipe  off  the 
grease  before  restoring  the  slide  to  its  place.  Too  much 
grease,  however,  causes  dust  to  adhere.  If  this  does 
not  remove  the  trouble,  a  little  grinding  with  finely 
powdered  pumice-stone  and  oil  may  help  the  difficulty; 
but  great  care  should  be  taken  to  wipe  off  all  abrading 
material.  Emery  in  any  form  should  never  be  used. 
If  a  slide  once  commences  to  fret,  it  rapidly  grows 
worse,  and  may  get  beyond  repair. 


CHAPTER  VII. 
THE  TRANSIT. 

ART.  1.     CONSTRUCTION. 

97.  THE  instrument  in  common  use  among  American 
engineers  for  measuring  horizontal  and  vertical  angles 
is  usually  called   a  transit.     Fig.  22  shows  the  general 
form.     It  is  sometimes,  but  incorrectly,  called  a  theodo- 
lite.    The  theodolite  is  the  name  given  by  British  en- 
gineers to  their  favorite  portable  instrument  (see  Fig. 
23,  page  93),  which  is  capable  of  performing  the  same 
work  as  the  transit. 

The  essential  difference  between  the  transit  and  the 
theodolite  is  that  in  the  former  the  telescope  can  transit 
or  turn  completely  over,  while  in  the  latter  it  can  not. 
The  telescope  of  the  theodolite  can  be  reversed  only  by 
lifting  it  out  of  its  supports  and  replacing  it  end  for  end, 
which  is  a  very  imperfect  substitute  for  the  revolution 
of  the  telescope  of  the  transit.  The  transit  is  sometimes 
called  an  engineer's  transit  and  sometimes  a  railroad 
transit,  to  distinguish  it  from  an  astronomical  transit. 

The  transit  was  invented  and  first  made  by  a  Phila- 
delphia firm  in  1831,  previous  to  which  time  the  English 
theodolite  and  the  magnetic  compass — sometimes  pro- 
vided with  a  full  circle  graduation,  by  which  angles 
could  be  read  independently  of  the  needle — were  the 
common  angle  instruments. 

98.  A  modern  engineer's  transit  has  more  than  350 
distinct  pieces.     Although  it  appears  quite  complicated, 
this  impression  disappears  when  each  part  is  examined 


THE    TRANSIT.  [CHAP.  Vll 


FIG.  22. — AMERICAN  TRANSIT. 


ART.   l]  CONSTRUCTION.  93 

in  turn,  and  its  uses  and  relations  to  the  rest  carefully 
studied.  The  great  value  of  the  transit  as  an  instru- 
ment of  precision  is  due  to  the  telescope,  by  which  great 
precision  in  sighting  is  attained,  and  to  the  graduated 
circle  with  its  vernier,  by  which  angles  can  be  read  with 
ease  and  accuracy.  All  other  parts  are  to  facilitate  the 
use  of  these  two. 


FIG.  23, — ENGLISH  THEODOLITE. 

The  tripod,  leveling  screws,  telescope,  and  vernier — 
all  very  important  parts  of  the  transit — have  been  dis- 
cussed in  preceding  chapters. 

99.  THE  GRADUATION.  The  lines  of  the  graduation 
should  be  uniform,  and  as  small  as  is  consistent  with 
legibility.  In  the  best  instruments  the  graduation  is 
upon  solid  silver. 

The  most  common  graduation  for  the  horizontal  limb 
of  transits  is  a  circle  about  6  inches  in  diameter  divided 
to  half-degrees  having  a  vernier  reading  to  minutes. 
The  better  transits  read  to  30",  and  some  to  20".  The 
degrees  are  usually  numbered  in  two  rows,  one  like  the 
compass  and  another  from  o°  to  360°.  The  field  work 
is  most  simple  and  least  liable  to  error,  if  only  the  latter 


94  THE    TRANSIT.  [CHAP.  VII 

numbering  is  used.  Or,  if  there  are  two  rows,  they 
should  be  like  those  shown  in  Fig.  13  (page  68). 

The  vertical  circle  should  be  numbered  from  o°  to 
90°;  for  then  angles  of  either  elevation  or  depression 
may  be  read  with  the  telescope  both  direct  and  inverted, 
and  the  mean  of  the  two  will  be  independent  of  errors 
of  adjustment  of  the  vertical  circle. 

100.  THE  VEBNIERS  on  the  horizontal  circle  are  some- 
times placed  90°  from  the  line  of  sight,  in  which  case 
the  observer  must  change  his  place  between  sighting 
the  telescope  and  reading  the  vernier.  Sometimes  the 
verniers  are  placed  immediately  under  the  telescope,  in 
which  case  the  observer  can  read  the  vernier  and  sight 
the  telescope  without  changing  his  position,  although 
the  telescope  must  be  revolved  before  the  vernier  can 
be  read.  The  latter  position  is  preferable,  especially 
in  confined  places,  as  in  underground  surveying,  etc. 
An  intermediate  position  would  be  still  better.  For 
convenience  of  reference,  the  verniers  should  have  some 
distinguishing  mark,  say  A,  B,  C,  etc.,  upon  their  faces. 

The  vernier  and  limb  should  be  exactly  in  the  same 
plane,  to  avoid  parallax  in  reading  ;  and  the  space  be- 
tween them  should  always  have  the  appearance  of  a 
uniform,  fine,  black  line.  The  verniers  should  be  pro- 
vided with  ground-glass  or  ivory  shades  for  illuminat- 
ing the  graduation. 

Verniers  reading  20  seconds  should  have  the  reading- 
glasses  permanently  attached  in  such  a  manner  that 
they  can  be  moved  radially  along  the  entire  length  of 
the  vernier.  The  tube  in  which  these  reading-glasses 
are  mounted  should  also  have,  at  the  end  nearest  the 
graduation,  a  fine  pointer  which  will  just  reach  the  end 
of  the  lines.  This  pointer,  being  in  the  center  of  the 
field,  will  serve  as  a  guide  in  moving  the  reading-glass 
exactly  opposite  the  coinciding  line.  The  center  of  the 
lens  is  thus  used  in  reading,  and  parallax  is  avoided. 


ART.   l] 


CONSTRUCTION. 


95 


101.  CENTERS,  OR  VERTICAL  AXES.  Usually  there  are 
two  concentric  vertical  axes,  the  verniers  and  telescope 
turning  about  the  inner,  and  the  graduated  circle  revolv- 
ing about  the  outer  one.  Fig.  24  shows  the  arrange- 
ment of  these  axes.  Ordinarily,  the  outer  axis  is  useful 
only  in  enabling  the  observer  to  shift  the  graduation 
so  as  to  begin  each  time  at  zero.  The  inner  axis  is 


FIG.  24. — SECTION  OF  THE  TRANSIT. 

made  more  carefully  than  the  outer.  These  two  axes 
are  often  called  "  centers "  and  sometimes  "compound 
centers."  For  work  not  requiring  great  accuracy,  but 
demanding  a  light  portable  instrument,  these  axes  are 
made  quite  short,  and  are  called  "  flat  centers."  The 
more  accurate  instruments  have  "  long  centers."  The 
instrument  should  turn  on  either  axis  without  the  slight- 
est play,  and  yet  with  very  little  friction. 

102.  LEVELS.  Since  a  transit  is  to  be  used  in  measur- 
ing horizontal  and  vertical  angles,  it  is  necessary  that 
levels  be  attached  to  the  instrument  to  determine  the 
plane  of  these  angles.  Two  level  vials,  perpendicular 


g6  THE    TRANSIT.  [CHAP.  VII 

to  each  other,  are  sufficient  to  bring  the  vertical  axis 
vertical,  which  then  serves  as  a  datum  to  which  to  ad- 
just the  other  parts  of  the  instrument.  If  the  instru- 
ment is  to  be  used  mainly  for  measuring  horizontal 
angles,  the  level  perpendicular  to  the  telescope  should 
be  the  more  sensitive.  In  ordinary  practice  vertical 
angles  are  seldom  required,  and  when  they  are  less 
precision  is  demanded  than  in  horizontal  ones;  hence 
it  is  very  common  to  attach  only  two  short  levels  to 
the  plate,  in  which  case  the  level  parallel  to  the  tele- 
scope should  be  the  more  sensitive.  If  the  instrument 
is  to  be  used  for  the  precise  measurement  of  vertical 
angles,  a  sensitive  level  should  be  attached  to  the  tele- 
scope to  indicate  a  horizontal  line — the  zero  for  vertical 
angles.  With  a  level  under  the  telescope,  a  transit  can 
be  used  as  a  leveling  instrument,  but  is  not  capable  of 
very  reliable  work,  owing  to  a  lack  of  stability. 

103.  CLAMP  AND  TANGENT  SCREW.*  With  the  unaided 
hand,  the  telescope  can  not  easily  be  made  to  cover  or 
bisect  the  exact  point  sighted  at;  and  to  assist  in  thus 
directing  the  telescope,  the  instrument  is  provided  with 
a  clamp,  and  a  tangent  or  slow-motion  screw. 

Clamps  are  made  in  a  variety  of  ways,  but  all  consist 
essentially  of  a  contrivance  by  which  a  piece  may  be 
connected  with  the  axis  or  rim  of  the  graduation  by 
tightening  a  screw.  The  clamp  is  connected  with  the 
vernier  plate  or  the  tripod,  as  the  case  may  be,  by  a 
screw  which  is  always  nearly  tangent  to  the  direction 
of  motion.  When  the  screw  is  turned,  the  two  parts  of 
the  instrument  rotate  slowly  with  reference  to  each 
other.  No  description  can  give  an  adequate  under- 
standing of  these  parts;  they  must  be  seen  and  examined 
to  be  comprehended. 

It  is  common  to  clamp  the  vernier  and  the  graduated 

*  Invented  by  Helvetius,  a  celebrated  astronomer  of  Dantzic,  about  1650, 


ART.   l]  CONSTRUCTION.  97 

plates  at  their  circumferences;  but  as  this  is  liable  to 
bend  the  plates,  it  is  far  better  to  clamp  the  axis  of  the 
plate. 

104.  A  perfect  tangent  screw  should  have  a  uniform 
motion,  and  be  free  from  lost  motion  or  "  back-lash,"  so 
as  to  respond  quickly  to  the  touch  and   yet  hold  the 
plate  and  vernier  exactly  as  set.     Lost  motion — the  com- 
mon defect  of  tangent  screws — is  a  source  of  great  an- 
noyance to  the  engineer,  and  often  causes  serious  errors 
in  the  field. 

The  tangent  screw  should  have  great  durability,  or  at 
least  an  even  wear,  so  that  the  screw  will  never  have  lost 
motion  in  one  part  and  move  hard  in  another.  The 
wear  is  greatly  lessened  by  covering  the  thread  with  a 
dust-guard,  as  is  now  quite  common.  The  tangent 
screw  should  be  so  made  as  to  work  at  least  fairly  well 
if  it  should  get  bent.  Several  forms  of  tangent  screws 
will  now  be  described. 

105.  English  Tangent  Movement.     The  oldest  and  most 
imperfect  of  all  is  that  known  as  the  English  or  stiff 
tangent  screw,  in  which  the    screw  works    through   a 
post  rn  the  clamp  and  against  a  collar  in  a  post  attached 
to  the  plate,  each  post  being  free  to  turn  about  a  verti- 
cal axis.     Fig.  25  shows  the  English  tangent  movement 


FIG,  25.— ENGLISH  TANGENT  MOVEMENT. 

as  applied  to  the  horizontal  circle  of  a  transit.  A  is  the 
head  of  the  clamp  screw  and  B  that  of  the  tangent 
screw,  This  form  is  defective  in  three  particulars:  i. 


98  THE    TRANSIT.  [CHAP.  VII 

Since  the  posts  are  movable  only  about  a  vertical  axis, 
if  the  screw  is  not  perfectly  straight  and  true  it  will 
bind  during  one  part  of  the  revolution.  2.  The  nut 
and  collars  should  be  the  same  height  above  the  plate; 
but  to  allow  for  errors  of  workmanship  in  satisfying  this 
condition,  the  hole  in  the  nut  is  often  made  too  large, 
and  the  nut  is  then  tightened  until  it  fits,  thus  causing 
it  to  touch  the  screw  along  only  two  lines,  and  it  there- 
fore soon  wears  loose.  3.  If  the  posts  are  tightened  up 
so  as  to  prevent  looseness,  the  friction  is  so  great  as  to 
interfere  with  the  freedom  of  their  motion. 

106.  Gambey  Tangent  Movement.  One  of  the  best 
forms  of  tangent  screws  is  the  Gambey  movement,  first 
made  by  the  celebrated  instrument-maker,  Gambey,  of 
Paris.  The  nut  is  a  split  sphere  which  may  be  confined 
like  any  ball-joint.  Instead  of  the  collar,  as  in  the 
English  form,  another  split  spherical  nut  which  works 
in  a  series  of  grooves  is  used.  This  form  of  tangent 
movement  is  shown  at  g,  Fig.  8,  page  59.  As  shown  in 
Fig.  8,  the  pieces  which  clamp  the  balls  are  too  heavy. 
They  should  be  made  light  enough  to  have  a  little 
flexibility  up  and  down,  thus  permitting  a  free  move- 
ment of  the  balls.  This  construction  provides  for  all 
necessary  motions  and  all  probable  imperfections  of  the 
screw. 

The  objections  to  this  form  are:  i.  The  balls  are  ex- 
posed to  flying  dust  and  sand,  which  tends  to  roughen 
their  surfaces  and  to  cause  an  uneven  motion.  Con- 
fining the  balls  between  springs  remedies  this  in  part. 
2.  As  ordinarily  made,  the  nut  is  much  shorter  than  the 
screw,  and  consequently  the  latter  wears  mainly  in  the 
middle,  and  after  a  little  use  the  screw  can  be  turned 
only  a  few  revolutions  without  having  lost  motion  in 
one  part  or  undue  friction  in  another.  This  defect  is 
removed  by  making  the  thread  in  the  nut  nearly  as 
Jong  as  the  thread  on  the  screw,  the  long  nut  being  con- 


ART.   l]  CONSTRUCTION.  99 

fined  as  the  ball  of  the  ordinary  form.  There  are  sev- 
eral slightly  different  ways  of  making  this  modification, 
all  of  which  are  very  satisfactory.  For  one  form  see 
Fig.  22,  page  92. 

107.  Spring  and  Abutting  Screw.     In  a  very  common, 
and  one   of    the    best,  forms    of    tangent    movement  a 
screw  abuts  against  the  clamp,  and  a  spring  on  the  op- 
posite side  keeps  the  clamp  in  contact  with   the  screw. 
To  prevent  excessive  local  wear  the  nut  through  which 
the  tangent  screw  works  should  be  long;  and  to  com- 
pensate for  wear  the  nut  should  be    made  adjustable 
and  have  a  little  elasticity.    An  objection  urged  against 
this  form  is  that  the  spring  causes  a  strain  between  the 
parts,  which  may  change  with   changes  of  temperature, 
and  which  is   liable  to  move  the  plates  or  derange  the 
levels.     This   objection    has   but   little    weight,  particu- 
larly if   there  is  some  means   of   tightening  the   spring. 
For  an  example  of  this  form  of  tangent  movement  see 
Fig.  28  (page  101)  and  also  Fig.  32  (page  131). 

108.  Two  Abutting  Screws.     Sometimes  two  opposing 
or  abutting  screws  are  used  to  get  rid  of  lost  motion; 
but  these  are  objectionable,  since  two   hands   must  be 
employed  in  using  them.     For  several  reasons  they  are 
not  at  all  suitable  for  the  upper  motion,  but  on  account 
of  their  steadiness  are  often   used  for  the  lower  motion. 
If  a  stiff  flat  spring  were  attached  to  the  side  of  the  lug 
against    which    the    tangent    screw    bears,   the    setting 
could  be  finished  with  one  screw.     Such  a  spring  is  oc- 
casionally so  added. 

109.  OBJECT-GLASS  SLIDE.     The  object-glass  should 

move  in  a  right  line  perpendicular  to  the  horizontal 
axis  of  the  telescope.  Some  instruments  have  an  ad- 
justment to  alter  the  direction  of  this  motion.  This  is 
accomplished  by  causing  the  inner  end  of  the  slide  to 
work  through  a  ring  which  is  adjustable  like  the  ring 
carrying  the  cross  hairs  (see  CC,  Fig.  27).  This  device 


JOO 


THE    TRANSIT. 


[CHAP,  vii 


FIG.  27.— SECTION  OF  TELE- 
SCOPE OF  TRANSIT. 


is  quite  objectionable,  since  the 
ring  is  liable  to  get  loose,  be- 
sides wearing  too  large.  Fur- 
thermore, the  adjustment  is 
quite  difficult  to  make,  and  no 
reason  can  be  assigned  why  the 
ring  should  not  be  fastened  per- 
manently when  once  in  the 
proper  position.  A  better  way 
is  to  make  the  whole  length  of 
the  slide  fit  the  inside  of  the 
telescope  tube.  The  latter 
method  requires  more  care  and 
greater  skill  in  the  manufacture. 
In  either  case  the  slide  should 
be  perfectly  straight. 

110.  GEADIENTEE.  This  is  a 
simple  device  which  adds  very 
much  to  the  convenience  and 
usefulness  of  the  transit.  It  is 
nothing  more  nor  less  than  an 
accurate  tangent  screw  with  a 
micrometer  head  working  beside 
a  scale  from  which  the  number 
of  complete  revolutions  is  as- 
certained.* It  can  be  applied 
to  the  vertical  or  horizontal 
limb,  and  may  be  used  in  estab- 
lishing grades,  determining 
horizontal  distances  (see  Art.  2, 
Chap.  X),  and  measuring  small 
angles  very  accurately  without 
an  arc  or  a  vernier. 


*  Invented    by    Professor     Stampfer,     of 
Vienna,  in  1873. 


ART.   l] 


CONSTRUCTION. 


IOI 


Fig.  28  shows  the  gradienter  as  applied  to  the  verti- 
cal circle  of  a  transit.  A  is  a  leg  attached  to  one  of  the 
standards  which  support  the  horizontal  axis  of  the 
telescope.  The  little  scale  immediately  above  the  grad- 
uated head  is  to  register  the  complete  revolutions 
of  the  screw,  the. fractions  of  a  revolution  being  read 
from  the  graduated  head.  The  screw  is  so  cut  that  one 
revolution  of  it  moves  the  telescope  through  an  angle 
whose  tangent  at  100  feet  from  the  instrument  is  i  foot 
or  0.5  foot.  In  the  former  case  the  head  is  divided 


FIG.  28. — GRADIENTER. 


into  one  hundred  parts,  and  in  the  latter  into  fifty; 
and  hence  in  either  case  a  unit  of  the  graduation  corre- 
sponds to  o.oi  of  a  foot  on  a  rod  100  feet  from  the 
instrument.  For  hints  on  the  use  of  the  gradienter  see 
§§  237-47- 


102  THE    TRANSIT.  [CHAP.  VII 

111.  SHITTING  PLATES.     This  is  a  device  for  bringing 
the    plummet     exactly    over     the    point,    after    having 
brought  it  approximately  into  position  by  manipulating 
the  tripod   legs   (§24).     The  usual  arrangement   allows 
the   whole   instrument    to  be  shifted    laterally  on    the 
tripod  head  about  an  inch.     This  is  a  very  great  con- 
venience. 

112.  COMPASS.     It  frequently  happens  that  it  is  abso- 
lutely necessary  to  run  lines  by  the  magnetic  compass, 
even    though    a    transit   is   at    hand  ;    and    therefore  a 
needle  and  compass  graduation  are   usually  added   to 
a  transit.     The    compass    is  also   valuable  as    a   check 
against   gross    errors    in    reading   the   vernier   of   the 
transit. 

The  tests  and  adjustments  of  the  compass  on  the 
transit  are  essentially  the  same  as  for  the  simple  com- 
pass (see  Chapter  III,  particularly  §  36). 

113.  VARIOUS  EXTRAS.     There  are  a  few  additions  and 
modifications  of  the  transit  which,  though  not  common, 
are  sometimes  made,  and  which  increase  the  range  and 
convenience    of    the    instrument    for   certain    kinds    of 
work.     A  few    of   the  most   important   of   these   extras 
will  now  be  described  briefly. 

When  it  is  desired  to  take  greater  vertical  angles 
than  is  possible  with  the  ordinary  eye-piece,  the  little 
cap  on  the  eye  end  of  the  telescope  is  unscrewed  and 
replaced  by  another  containing  a  small  prism  which 
reflects  the  beam  at  right  angles  and  brings  it  out  to 
the  eye  of  the  observer.  When  the  telescope  is  used  to 
look  at  the  sun,  a  colored  shade  must  be  placed  over 
the  object-glass  or  be  interposed  between  the  eye  and 
the  telescope.  A  piece  of  smoked  window-glass,  held 
in  the  hand,  will  serve  the  purpose;  but  caps  contain- 
ing colored  shades  are  made  by  the  various  instrument- 
manufacturers,  which  are  much  more  convenient,  and 
Comparatively  inexpensive. 


ART.   2]  TESTING    THE    TRANSIT.  IOJ 

When  the  telescope  is  used  in  a  mine,  or  to  look  at 
a  star,  unless  some  diffused  light  enters  the  telescope, 
the  cross  hairs  will  not  be  visible.  Caps  are  made 
which  support  an  annulus  in  front  of  the  objective,  at 
an  angle  of  45°  with  the  telescope;  and  the  cross  hairs 
are  then  illumined  by  holding  a  lamp  or  candle  at  the 
side  of  the  objective.  The  same  thing  may  be  accom- 
plished, though  less  easily,  with  a  piece  of  paper  held  in 
the  hand  or  fastened  to  a  board. 

In  mine  surveying  it  is  often  necessary  to  sight  verti- 
cally up  or  down  a  shaft.  For  this  purpose  an  extra 
telescope  is  sometimes  attached  to  the  end  of  the  hori- 
zontal axis;  or  sometimes  the  standards  are  inclined, 
so  that  the  ordinary  telescope  may  sight  vertically 
downward  past  the  plates.  See  §  168  for  still  another 
method. 


ART.  2.     TESTING  THE  TRANSIT. 

114.  GRADUATION.  The  graduation  may  have  two 
classes  of  errors — accidental  and  periodic.  Accidental 
errors  are  those  which  follow  no  regular  law,  and  are 
equally  liable  to  occur  at  any  given  division.  There  is  a 
variety  of  causes  which  may  produce  them.  Periodic 
errors  are  those  which  follow  some  law,  and  are  proba- 
bly caused  by  some  peculiarity  of  the  graduating  en- 
gine. 

Since  the  accuracy  of  the  graduation  is  a  vital  point, 
it  is  very  unfortunate  that  there  is  no  easy  or  simple 
method  of  testing  it.  An  imperfect  test  is  to  notice 
whether  the  extreme  lines  of  the  vernier  span  the  same 
number  of  divisions  in  all  parts  of  the  circle.  But  with 
the  present  graduating  engines  it  is  very  poor  gradua- 
tion indeed  whose  errors  may  be  detected  in  this  way. 

With  astronomical  and  geodetic  instruments  very 
elaborate  methods  are  employed  for  detecting  errors  of 


IO4  THE    TRANSIT.  [CHAP.  VII 

graduation;  but  these  methods  are  too  complicated  to 
be  explained  here,  and,  moreover,  they  are  not  applicable 
to  common  engineering  instruments. 

115.  ECCENTRICITY.     The  center  of  graduation  should 
lie   in    the   axis   of   rotation.     To    test    this,    read   two 
verniers    180°    apart    (or    any    number   of    equidistant 
verniers),  then  move  the  circle  preferably  90°  and  read 
again.     If  the  verniers  have  changed  the  same  amount, 
the    circle    is   well   centered.      If    the    two   have     not 
changed    the    same    amount,    the     mean    of    the    two 
differences   is  the  actual  angle  through  which   the  in- 
strument has   been   moved.     Eccentricity  is,  therefore, 
wholly   eliminated    by    reading    two    opposite    verniers 
each  time,  and  taking  the  mean.     The  horizontal  circle 
of  the  common  engineer's  transit  seldom  has  more  than 
one  minute  of  eccentricity.     There  is  usually  but  one 
vernier  on  the  vertical  limb,  and  hence  there  is  no  way 
of  testing  or  eliminating  its  eccentricity. 

It  is  frequently  said  that  if  the  readings  of  the  two 
verniers  differ  by  180°,  the  graduation  and  centering 
are  perfect;  but  this  is  no  criterion.  The  difference 
between  the  verniers  depends  upon  the  accuracy  of 
the  graduation,  the  eccentricity,  and  the  angular  dis- 
tance between  verniers.  The  readings  might  differ  by 
180°,  and  still  the  graduation  and  centering  be  very 
much  in  error. 

Assuming  the  graduation  to  be  perfect,  the  difference 
between  the  two  verniers  can  be  expressed  by  180  ±  €• 
If  e  remains  constant,  the  centers  coincide,  but  the 
verniers  are  not  opposite;  if  e  varies,  the  verniers  are 
opposite,  but  the  centers  do  not  coincide. 

116.  MAGNIFYING  POWER  vs.  VERNIER.    The  magnify- 
ing power  of  the  telescope  and  the  least  count  of  the 
vernier  should  be  so  proportioned  that   the   least  per- 
ceptible movement  of  the  vernier  will  cause  sufficient 
movement  of  the  cross  hairs  on  the  object  to  be  easily 


ART.   2J  TESTING    THE    TRANSIT. 


noticed  through  the  telescope;  and,  vice  versa,  the  least 
noticeable  motion  of  the  cross  hairs  on  the  object 
should  cause  a  barely  perceptible  change  of  the  vernier. 
Since  the  horizontal  circle  is  usually  (and  properly)  the 
more  accurate,  this  condition  applies  more  especially  to 
the  horizontal  circle.  A  higher  power,  or  a  smaller 
count  of  the  vernier,  than  that  required  by  this  condi- 
tion is  detrimental,  the  former  causing  an  unnecessary 
loss  of  light  and  the  latter  a  waste  of  time  in  reading 
the  vernier. 

117.  MAGNIFYING  POWER  vs.  LEVEL  UNDER  TELESCOPE. 

If  the  transit  is  to  be  used  as  a  leveling  instrument,  the 
magnifying  power  of  the  telescope  and  the  delicacy  of 
the  level  under  the  telescope  should  be  so  proportioned 
that  a  barely  perceptible  movement  of  the  cross  hairs 
on  the  object  will  cause  a  just  perceptible  movement  of 
the  bubble. 

118.  MAGNIFYING  POWER  vs.  PLATE  LEVELS.    The  re- 
lation   which    should    exist    between    the    magnifying 
power  of  the  telescope  and  the  sensitiveness  of  the  plate 
levels  depends  wholly  upon  the  kind  of  work  to  be  done. 
The  following  method  of  testing  this  relation   suffices 
for   all    kinds    of   work.       Bring   the    bubbles   of   both 
plate  levels  to  the  middle,  and  sight  upon  a  well-defined 
point  at  as  great  an  angular  elevation  as  the  instrument 
will  permit ;  then  slightly  derange  the  levels  by  manip- 
ulating the  foot  screws,  and  carefully  level  up  again. 
If,  on   again   looking   through   the  telescope,   the  cross 
hairs  cover  the  same  point  as  before,  the   sensitiveness 
of  the  levels  is  proportional  to  the  magnifying  power  of 
the  telescope.     If  the  cross  hairs  are  out  horizontally, 
it  shows  that  the  level  perpendicular  to  the  telescope  is 
not  sensitive  enough.     If  the  cross  hairs  are   out  verti- 
cally, the  level  parallel  to  the  telescope  is  too  sluggish. 
This  relation  is  much  less  important  than  the  two  pre- 
ceding ones. 


I06  THE    TRANSIT.  [CHAP.  VII 

119.  PARALLELISM  OF  VERTICAL  AXES.    The  vertical 

axes  should  not  only  be  parallel  to  each  other,  but 
should  also  be  concentric.  To  test  the  first  condition, 
adjust  the  most  sensitive  level  about  the  instrument 
perpendicular  to  one  axis  (§  38),  clamp  that  axis,  and 
revolve  the  instrument  about  the  other;  then  if  the 
Bevels  are  in  adjustment  about  the  second  axis  also,  the 
axes  are  parallel.  If  the  axes  are  not  parallel,  no  error 
will  be  produced  except  when  the  instrument  is  used  to 
measure  angles  by  repetition  (§  132),  provided  the  plate 
levels  are  adjusted  perpendicular  to  the  irmer  axis. 

120.  LIMB  PERPENDICULAR  TO  Axis.    The  plane  of  the 
graduation  should  be  perpendicular  to  the  axis  about 
which  it  revolves.     If  the  limb  is  not  horizontal,  angles 
measured  on  one  part  of  it  will   be   too  great,  and  an- 
gles 90°  therefrom  will  be  as  much  too   small.     How- 
ever, the  error  would  almost  certainly  be  inappreciable 
with  common  engineering  instruments. 

If  desired,  this  condition  can  be  tested  as  follows: 
Bring  the  vertical  axis  vertical  (§  38),  place  a  block 
level  on  the  horizontal  limb,  read  the  position  of  the 
bubble  and  reverse  the  limb  ;  then  reverse  the  block 
level  to  eliminate  its  error,  and  read  the  position  of  the 
bubble  again.  If  the  bubble  reads  alike  both  times, 
the  horizontal  limb  is  perpendicular  to  the  vertical  axis. 

121.  OBJECT-GLASS  SLIDE.     The  optical  center*  of  the 
objective  should  be  projected  in  the  line  of  collimation.f 

*  The  optical  center  is  a  point  so  situated  that  any  ray  of  light  passing 
through  it  will  undergo  equal  and  opposite  refractions  on  entering  and  leav- 
ing the  lens.  It  will,  therefore,  be  found  where  a  line  joining  the  extremities 
of  two  parallel  radii  of  the  opposite  surfaces  intersects  the  optical  axis  of  the 
lens.  For  a  double-convex  lens,  it  is  always  within  the  surface  of  the  lens. 
For  a  plano-convex  or  a  plano-concave  lens,  the  optical  center  will  be  at  the 
intersection  of  the  axis  with  the  curved  surface. 

A  lens  has  been  defined  as  a  mathematical  point  which  allows  a  great  deal 
of  light  to  go  through  it.  The  better  the  lens  the  more  nearly  is  this  con- 
dition realized.  This  "  point"  is  the  optical  center. 

t  The  line  joining  the  intersection  of  the  cross  hairs  and  the  optical  center. 


ART.  2]  TESTING    THE    TRANSIT.  1O7 

If  it  is  not  so  projected,  and  the  instrument  be  colli- 
mated  for  one  distance  (§  123),  it  will  be  out  of  colli- 
mation  for  every  other  distance.  This  is  equivalent  to 
requiring  that  the  slide  shall  be  straight  and  move  in 
the  plane  of  the  horizontal  axis  and  perpendicular  to  it. 
If  the  objective  is  fixed  and  the  cross  hairs  are  mova- 
ble, the  same  conditions  should  be  satisfied,  and  the 
method  of  testing  is  the  same  in  both  cases.  To  make 
this  test,  collimate  the  instrument  (§  123,  or  §  278,  or 
§  285)  and  test  (i)  for  a  deviation  from  a  vertical  plane, 
(2)  for  a  deviation  from  a  plane  at  right  angles  to  the 
vertical  plane,  (3)  for  a  deviation  from  the  perpendic- 
ular to  the  horizontal  axis,  and  (4)  for  a  deviation  from 
the  plane  of  the  horizontal  axis. 

i.  To  test  for  a  deviation  from  a  vertical  plane.  Se- 
lect smooth  ground  and  lay  off  any  number  of  points, — 
say  five  or  six,  100  feet  apart, — having  one  as  near  the 
instrument  as  possible,  and  line  the  points  carefully 
with  the  telescope.  Reverse  in  altitude  and  azimuth, 
sight  upon  the  first  point,  and  clamp  the  instrument ; 
then  locate  points  in  line  with  the  first  one,  near  each 
of  the  others.  Measure  the  distances  between  the  sev- 
eral pairs  of  points.  If  these  distances  vary  as  the  dis- 
tance from  the  first  point,  the  slide  is  probably  straight 
and  the  optical  center  is  projected  in  a  vertical  plane 
but  not  perpendicular  to  the  horizontal  axis.  To  cause 
the  optical  center  to  move  in  a  plane  perpendicular  to 
the  horizontal  axis,  sight  upon  the  first  point  and  clamp 
the  vertical  axis  ;  then  direct  the  telescope  to  the  second 
point,  and  move  the  back  end  of  the  slide  to  correct 
half  the  error.  Moving  the  back  end  of  the  objective 
slide  may  have  destroyed  the  adjustment  for  collima- 
tion  ;  therefore  recollimate  the  instrument,  and  again 
test  for  straightness  of  slide.  If  the  instrument  can  not 
be  adjusted  to  bisect  a  row  of  points  before  and  after 
reversal,  the  slide  is  not  straight,  and  no  good  work 


108  THE    TRANSIT.  [CHAP.  VII 

can  be  done  with  the  instrument.  Of  course,  due  con- 
sideration must  be  given  to  the  errors  of  observation. 
An  error  may  be  introduced  by  the  slide's  being  loose  ; 
but  this  may  be  avoided  by  finishing  every  setting  of 
the  object-glass  by  motion  always  in  the  same  direction, 

2.  To  test  for  a  deviation  from  a  plane  at  right  angles  fo 
a  vertical  plane.    The  method  is  similar  to  the  preceding. 
Drive  several  stakes  into  the  ground  at  equal  intervals, 
the  first  being  near  the  instrument.     Read  a  leveling 
rod  upon  each,  reverse  in  altitude   and    azimuth,   and 
bring   the   line  of  sight  to  the   previous  reading  upon 
the  first  stake;   then  read  the  rod  upon  all  the  other 
stakes.     If  the  differences  of  readings  vary  as  the  dis- 
tance  from    the   first   station,  the  object-glass   is    pro- 
jected in  a  plane  perpendicular  to  the  vertical   plane. 
Since  it  is  projected  in  two  planes  at  right  angles  to 
each  other,  it  must  be  in  their  intersection;  therefore 
the  optical  center  is  projected  in  a  straight  line,  and  the 
slide  is  straight. 

3.  To  test  whether  the  motion  is  perpendicular  to  the  hori- 
zontal axis.  Collimate  the  vertical  hair  (i  §  123)  for  a  near 
distance;  then,  if  it  is  in  collimation  for  a  greater  dis- 
tance, the  optical  center  of  the  objective  is  projected  in 
a  line  perpendicular  to  the  horizontal  axis  of  the  tele- 
scope.    If  it  is  not  in  collimation  for  the  farther  point^ 
move  the  back  end  of  the  object-glass  slide  until  it  is; 
but  if  there  is  no  means  of  adjusting  the  slide,  send  the 
instrument  to  the  maker.     See  §  109. 

4.  The  rigorous  condition  is  that  the  line  of  motion  of 
the  optical  center  should  pass  through  the  horizontal  axis.     It 
would  be  impossible  to  certainly  secure  this  relation; 
but  for  any  cases  likely  to  occur  in   practice,  including 
those  requiring  astronomical  accuracy,  it  is  sufficient  to 
require  that  the  line    of    collimation  be  in  the  line  of 
motion    of    the    optical    center.     Therefore    it    is    only 
necessary  to  place  the   horizontal   hair  in   the   line    of 


ART.  3]  ADJUSTMENTS    OP    THE    TRANSIT.  109 

motion  of  the  optical  center,  which  is  an  adjustment, 
and  will  be  discussed  in  Art.  3  of  this  chapter. 

ART.  3.     ADJUSTMENTS  OF  THE  TRANSIT.* 

122.  LEVELS.     The  levels  should  be   perpendicular  to 
the  vertical  axis,     This  is  the  same  adjustment  as  that 
described   for  the  compass   (§§  38  and  39,  which  see). 
As  in  the  compass,  this  condition  is  important  only  in 
certain  kinds  of  work,  as  will  be  discussed  in  §  128. 

123.  CROSS   HAIRS.     This    adjustment    is    ordinarily 
known  as  the  adjustment  of  the  line  of  collimation.f 
For  obvious  reasons,  it  is  better  to  call  it  the  adjust- 
ment of  the  cross  hairs.     It  is  made  in  two  steps,  (i) 
the  adjustment  of  the  vertical  hair,  and  (2)  that  of  the 
horizontal  hair. 

1.  Vertical  Hair.  The  line  of  collimation  should  be 
perpendicular  to  the  horizontal  axis  of  the  telescope. 
Notice  that  if  this  condition  is  satisfied,  the  line  of  colli- 
mation will  describe  a  plane  during  the  revolution  of 
the  telescope;  but  if  it  is  not,  it  will  describe  the  surface 
of  a  cone.  To  make  this  adjustment,  level  the  instru- 
ment, and  sight  upon  some  well-defined  point;  then 
reverse  the  instrument  in  altitude,  and  fix  a  point  in 
line  at  an  equal  distance  from  the  instrument.  (If  the 
standards  have  not  been  adjusted,  or  if  the  horizontal 
axis  is  not  level,  these  points  must  be  in  the  same  hori- 
zontal line,  or  nearly  so.)  Reverse  the  instrument  in 
azimuth  and  sight  at  the  first  point;  then  reverse  in 
altitude,  and  mark  a  point  in  line  at  the  same  distance 
as  before.  If  the  two  points  do  not  agree,  move  the 
vertical  hair  one  fourth  of  the  difference,  and  repeat  the 
whole  operation  to  test  the  accuracy  of  the  adjustment. 

*  For  general  remarks  upon  adjustments,  see  §  37. 

t  The  line  joining  the  intersection  of  the  cross  hairs  and  the  optical  center 
of  the  objective  is  the  line  of  collimation. 


HO  THE    TRANSIT.  [CHAP.  VII 

If  there  is  any  back-lash  in  the  tangent  screw,  the 
instrument  must  be  manipulated  very  carefully,  to  pre- 
vent any  change  of  position  on  the  vertical  axis.  There 
are  several  methods  of  making  this  adjustment,  but 
this  is  the  most  simple,  as  well  as  the  most  accurate. 
Notice  that  this  method  is  independent  of  all  instru- 
mental errors,  with  the  possible  exception  that  the  line 
of  collimation  does  not  lie  in  the  plane  of  the  vertical 
axis;  but  any  possible  error  in  this  respect  will  be 
inapppreciable  except  at  very  short  distances. 

Next  proceed  to  make  the  vertical  hair  vertical.  This 
adjustment  is  useful  only  in  sighting  at  a  flag-pole,  to 
tell  whether  it  is  vertical.  The  simplest  way  to  make 
this  adjustment  is  to  level  the  horizontal  axis  of  the 
telescope  (§  126),  and  see  whether  the  hairs  will  coin- 
cide with  the  corner  of  some  building.  Or,  move 
rigorously,  having  the  axis  of  the  telescope  horizontal, 
sight  upon  some  well-defined  point,  and  elevate  or 
depress  the  telescope.  If  the  hair  is  vertical,  the  point 
will  seem  to  travel  up  and  down  the  hair;  if  it  does 
not,  loosen  the  screws  a  trifle  and  turn  the  diaphragm 
slightly. 

Before  pronouncing  upon  any  adjustment,  repeat  the 
test  to  discover  the  error  of  observation.  The  greater 
the  accuracy  of  the  instrument,  the  more  important  this 
precaution. 

2.  Horizontal  Hair.  The  horizontal  hair  should  be 
in  the  plane  of  motion  of  the  optical  center  of  the  ob- 
jective. If  this  condition  is  not  satisfied,  the  line  of 
collimation  will  change  with  every  change  of  the  object- 
glass,  rendering  the  instrument  useless  for  leveling,  or 
for  measuring  vertical  angles.  In  discussions  of  the 
adjustments  of  the  transit  no  mention  is  ever  made  of 
an  adjustment  of  the  horizontal  hair;  although  such 
an  adjustment  is  necessary  if  the  instrument  is  to  be 


ART.  3]  ADJUSTMENTS    OF    THE    TRANSIT.  Ill 

used  for  anything  except  the  measurement  of  hori- 
zontal angles. 

To  make  this  adjustment,  drive  a  stake  near  the  in- 
strument and  read  a  level-rod  upon  it;  then,  without 
moving  the  telescope  in  altitude,  read  a  rod  upon  a 
second  stake  200  or  300  feet  distant.  Reverse  in  alti- 
tude and  azimuth,  and  bring  the  telescope  to  the 
former  reading  upon  the  first  point;  then  read  upon  the 
other  stake.  If  this  reading  is  not  the  same  as  before, 
correct  half  the  difference  by  moving  the  horizontal 
hair.  After  having  adjusted  the  horizontal  hair,  test 
the  vertical  hair,  for  moving  one  may  have  changed  the 
other. 

It  would  be  a  great  improvement  if  telescopes  were 
so  made  as  not  to  require  an  adjustment  of  the  hori- 
zontal hair  of  the  transits  or  vertical  hair  of  levels. 
This  improvement  could  be  most  easily  applied  to  in- 
verting telescopes,  and  is  another  reason  for  using  such 
telescopes.  In  case  the  hair  is  non-adjustable,  the  en- 
gineer should  test  the  adjustment  for  himself. 

124.  CENTERING   THE   EYE-PIECE.    After  having  ad- 
justed the  cross  hairs,  it  may  be  that  their  intersection 
will  not  be  in  the  middle  of  the  field  of  view.      This 
does  not  affect  the  accuracy  of  the  work,  but  does  affect 
the  seeing  power  of  the  telescope.     Some  instruments 
are  provided  with  two  pairs  of  screws  (A  A,  Fig.  27, 
page  100),  like  the  screws  which  move  the  cross-hair 
ring,  by  which  the   inner  end  of  the  eye-piece  may  be 
moved  so  that  the  cross  hairs  shall  appear  in  the  center 
of  the  field.     In  inverting  telescopes,  no  means  is  pro- 
vided for  making  this  adjustment. 

125.  STANDARDS.     The  line  of  collimation  should  re- 
volve in  a  vertical  plane  when  the  vertical  axis  is  verti- 
cal;  or,  in  other  words,  the  horizontal  axis  should   be 
horizontal   when   the  instrument  is  leveled    up,  i.e.,  the 
standards  should  be  of  the  same  length. 


112  THE    TRANSIT.  [CHAP.  VII 

To  make  this  adjustment,  having  adjusted  the  cross 
hairs  (§  123),  level  the  instrument  very  carefully,  direct 
the  telescope  to  some  high  and  well-defined  point,  and 
clamp  the  vertical  axis.  Establish  a  low  point  in  the 
line  of  the  telescope  ;  reverse  in  altitude  and  azimuth, 
direct  the  telescope  to  the  high  point,  and  clamp  the 
vertical  axis.  If  the  telescope,  when  turned  down,  does 
not  cover  the  low  point,  correct  one  half  the  difference 
by  lowering  the  end  of  the  axis  towards  which  the  line 
of  sight  diverges. 

In  making  this  adjustment,  it  is  only  necessary  to 
level  the  bubble  perpendicular  to  the  telescope.  It  is 
sometimes  desirable  to  throw  the  instrument  into  a  po- 
sition that  will  command  a  large  vertical  angle,  when 
the  level  parallel  to  the  telescope  can  not  be  used. 
If  the  line  of  collimation  has  not  been  adjusted  previ- 
ously, the  points  used  must  be  equally  above  and  below 
the  instrument,  and  at  the  same  distance  from  it;  for 
it  must  be  borne  in  mind  that  if  the  line  of  collimation 
is  not  perpendicular  to  the  axis  of  the  telescope,  it  de- 
scribes the  surface  of  a  cone. 

Another  method  of  making  this  adjustment  is  to 
cause  the  line  of  sight  to  follow  a  plumb-line  ;  or,  what 
is  in  effect  the  same  thing,  require  it  to  cover  a  high 
point  and  its  reflection  as  seen  in  a  basin  of  mercury 
or  water.  Of  these  two,  the  second  is  the  better,  owing 
to  the  vibrations  of  the  plumb-line  ;  but  both  are  really 
tests  of  the  accuracy  of  the  adjustment  of  the  level  as 
well  as  of  the  standards,  and  therefore  neither  of  them 
is  as  good  as  the  one  first  described. 

126.  LEVEL  UNDER  TELESCOPE.  The  tangent  of  the 
level  should  be  parallel  to  the  line  of  collimation  when 
the  latter  is  horizontal.  To  make  this  adjustment, 
bring  the  bubble  to  the  middle  (any  point  will  do 
equally  well,  but  the  middle  is  most  convenient),  and 
clamp  the  telescope.  Read  a  leveling-rod  on  a  point, 


ART.  3]  ADJUSTMENTS    OF    THE    TRANSIT. 


say  200  feet  from  the  instrument  ;  reverse  in  azimuth, 
bring  the  bubble  to  its  former  position  by  moving  the 
tangent  screw  of  the  vertical  circle,  and  establish  a 
second  point  at  the  same  distance  from  the  instrument 
as  the  first  one,  and  read  the  rod  upon  it.  A  line  join- 
ing the  two  positions  of  the  target  is  a  horizontal  line, 
and  the  difference  of  the  readings  is  the  true  difference 
of  level,  however  much  the  instrument  may  be  out  of 
adjustment.  Move  the  instrument  very  near  to  one 
point — say,  10  feet  beyond  it, — and  read  a  rod  upon  the 
first  point  ;  then,  without  changing  the  altitude  of  the 
telescope,  read  upon  the  second.  If  the  difference  of 
these  two  readings  is  the  same  as  the  difference  of  level, 
the  line  of  collimation  is  horizontal.  If  the  observed 
difference  is  not  equal  to  the  difference  of  level,  correct 
a  little  more  than  all  the  error  (for  the  exact  amount 
see  the  next  paragraph)  by  moving  the  farther  target. 

Let  A  and  B,  Fig.  29,  be  the  two  points.     When  the 
instrument  is  at  C,  midway  between  A  and  B,  the  tar- 


FIG.  29. 

gets  are  at  a  and  b,  and  ab  is  a  level  line  ;  and  the  true 
difference  of  level  between  A  and  B  =  Aa  —  Bb.  When 
the  instrument  is  at  Z>,  the  targets  are  at  c  and  d\  and 
the  apparent  difference  of  level  is  de  =  (Be  —  Bb)  — 
(Ad  —  Aa).  If  the  line  of  sight  were  horizontal,  the 
target  would  be  at/;  and  therefore  df  is  the  correction. 
AD  _  ^  (zAC+BD\  * 


a) 


*  This  formula  is  not  strictly  correct,  since  (i)  it  was  assumed  that  a  line 
which  is  horizontal  at  C,  i.e.,  perpendicular  to  the  radius  of  the  earth  at  C,  is 


1 14  THE    TRANSIT.  [CHAP.  VII 

If  D  is  between  A  and  B,  the  quantity  BD  must  be 
subtracted. 

In  making  this  adjustment,  take  two  or  three  pairs 
of  observations  each  time  for  a  check,  and  be  careful 
to  see  that  the  instrument  is  exactly  level  at  the  mo- 
ment of  sighting. 

When  the  line  of  sight  is  brought  horizontal,  bring 
the  bubble  to  the  middle  by  raising  one  end  of  the 
level  tube,  and  the  adjustment  is  complete.  In  moving 
the  level  tube,  be  very  careful  not  to  alter  the  inclina- 
tion of  the  telescope.  Notice  that  it  will  not  do  to 
move  the  horizontal  cross  hair  to  bring  the  line  of  colli- 
mation  parallel  to  the  level,  as  is  sometimes  recom- 
mended, for  moving  the  cross  hair  will  destroy  its  ad' 
justment  for  collimation  (2,  §  123),  and  may  destroy 
that  of  the  vertical  hair. 

127.  ZERO  OF  THE  VERTICAL  CIRCLE.  The  vertical 
circle  should  read  zero  when  the  vertical  axis  is  vertical 
and  the  line  of  sight  is  horizontal.  If  the  instrument 
has  a  level  under  the  telescope,  first  adjust  it  (§  126)  ; 
then  bring  the  bubble  to  the  middle,  and  shift  the 
vernier  until  its  zero  coincides  with  the  zero  of  the 
limb.  If  the  vernier  is  not  movable,  note  the  difference 
and  apply  it  as  a  correction  to  each  angle  of  elevation 
or  depression.  Notice  that  this  adjustment  is  not  nec- 
essary if  only  the  vertical  angle  between  two  points  is 
desired. 

If  the  instrument  has  no  level  under  the  telescope, 
the  method  of  making  this  adjustment  is  nearly  the 


parallel  to  a  line  which  is  horizontal  at  Z>,  and  (2)  the  effect  of  refraction  was 
omitted.  Therefore  the  observations  at  D  should  be  corrected  for  curvature 
and  refraction  (§§  317-19).  To  make  this  correction,  subtract  o.oc*(BD  -H  2oo)2 
ft.  from  the  reading  at  B,  and  o.ooi[(2  AC -\-  BD)  -4-  220]*  ft.  from  that  at 
A\  or,  with  sufficient  accuracy,  simply  subtract  0.001(2  A  C  -s-  220) *  ft.  from 
de  before  using  it  in  equation  (i).  Ordinarily  this  correction  is  less  than  the 
error  of  observation,  and  consequently  may  generally  be  omitted. 


ART.  4]  USING    THE    TRANSIT.  115 

same  as  that  given  in  §  126.  Set  the  instrument  mid- 
way between  two  points,  level  it  carefully,  taking  special 
care  with  the  level  parallel  to  the  vertical  circle,  clamp  the 
telescope  nearly  horizontal,  and  determine  the  differ- 
ence of  level  between  the  points.  Move  the  instrument 
near  one  of  the  points,  level  it  carefully,  sight  upon 
each  point,  and  take  the  difference  of  reading.  If  this 
difference  is  equal  to  the  true  difference  of  level,  as 
found  when  the  instrument  was  midway  between  the 
points,  the  line  of  sight  is  horizontal.  If  the  differ- 
ence is  not  equal  to  the  difference  of  level,  alter  the 
telescope  in  altitude  (see  the  second  paragraph  of  §  126) 
until  the  differences  are  the  same.  Finally,  when  the 
line  of  sight  has  been  placed  horizontal,  move  the  ver- 
nier to  coincide  with  the  zero  of  the  circle. 

ART.  4.     USING  THE  TRANSIT. 

128.  PRACTICAL  HINTS.  The  beginner  should  not 
neglect  the  preliminary  matters  of  planting  the  tripod, 
bringing  the  plumb-bob  over  the  point,  and  leveling 
the  instrument.  There  is  great  difference  in  the  skill 
and  rapidity  with  which  different  persons  will  set  up 
the  transit;  but  generally  there  is  an  unnecessary 
waste  of  time  and  hard  labor.  A  very  neat  way  of 
doing  it  is  as  follows  :  Tighten  the  screws  in  the  upper 
end  of  the  legs  until  friction  will  just  hold  them  wher- 
ever placed,  and  open  them  until  they  make  an  angle 
of  about  30°  with  the  vertical,  let  down  the  plumb-bob, 
and  set  the  instrument  over  the  point  ;  then  a  gentle 
pressure  on  the  legs  of  the  tripod  will  bring  the  plum- 
met over  the  point,  and  a  slight  movement  of  the  level- 
ing screws  will  bring  the  plate  level. 

If  the  observer  has  clearly  in  mind  what  he  is  trying 
to  do,  and  thoroughly  comprehends  the  effect  of  possi- 
ble errors  in  his  instrument,  he  can  save  much  time  and 


Il6  THE    TRANSIT.  [CHAP.  VII 

trouble,  thus  leaving  himself  free  to  bestow  his  atten- 
tion where  it  will  do  the  most  good.  For  example,  a 
great  deal  of  time  is  often  wasted  in  getting  the  instru- 
ment precisely  over  the  point.  The  care  should  vary 
inversely  as  the  distance  of  the  object  sighted  at  ;  an 
inch  may  cause  an  error  of  three  seconds  if  the  object 
is  a  mile  away,  but  an  error  of  three  minutes  if  100  feet 
away.  The  direction  of  the  instrument  from  the  point, 
with  reference  to  the  object  sighted  at,  also  affects  the 
value  of  the  resulting  error. 

It  is  a  waste  of  time  to  level  up  accurately,  each  time, 
regardless  of  the  kind  of  work  to  be  done.  If  only  the 
horizontal  angles  between  points  on  the  same  level  are 
to  be  found,  the  instrument  can  be  leveled  with  suffi- 
cient accuracy  by  the  eye  alone ;  if  only  vertical  angles 
between  points  in  the  same  vertical  are  desired,  only 
the  level  parallel  to  the  telescope  need  be  read.  On 
the  other  hand,  in  measuring  a  horizontal  angle  be- 
tween a  high  and  a  low  point,  particular  attention 
should  be  given  to  the  level  perpendicular  to  the  tele- 
scope. In  all  instrumental  work  there  are  certain  oper- 
ations which  should  be  carefully  attended  to,  while  to 
give  equal  care  to  others  would  be  only  a  waste  of  effort. 

If  the  instrument  is  not  firm,  examine  the  tripod- 
head  and  the  iron  shoes  on  the  legs,  to  see  that  they 
are  not  loose.  No  instrument  can  stand  firmly  with  any 
looseness  in  these  parts.  The  clamps  and  tangent 
screws  also  should  be  examined  to  see  that  they  fit 
snugly.  The  instrument  may  slip  on  the  lower  plate, 
owing  to  the  leveling  screws'  not  being  tight  enough. 

Some  engineers  seem  to  think  that  the  harder  the 
tripod  legs  are  forced  into  the  ground,  the  tighter  the 
leveling  screws  are,  and  the  tighter  the  instrument  is 
clamped,  the  more  accurate  the  work  ;  but  the  contrary  is 
more  nearly  true.  An  instrument  keeps  its  adjustments 
better  and  works  more  kindly,  when  handled  delicately. 


ART,  4]  USING    THE    TRANSIT.  II 7 

129.  MEASURING   ANGLES.     Although  it  is    scarcely 
necessary,  a  brief  description  of  the  method  of  measur- 
ing a  horizontal  angle  will  now  be  given.     Set  the  in- 
strument over  the  point,  and  level  it.     Then  clamp  the 
upper  movement,  and  read  one  of  the  verniers,  say  A, 
using    the   o°-36o°    graduation.       Focus    the   eye-piece 
on  the  cross  hairs,  turn  the  telescope  by  hand  until  it 
nearly  bisects  one  of  the  points,  and  clamp   the  lower 
motion.     Next  focus  on  the  object  and  turn  the  lower 
tangent  screw  until  the  intersection  of  the  cross  hairs 
exactly  covers    the    point.     Then  loosen  the  clamp  of 
the    upper   motion,   direct   the   telescope  to   the  other 
point,  and  read  as  before.     The   difference  of  the  two 
readings  will  be  the  angle  between  the  two  points. 

It  is  immaterial  whether  the  instrument  is  turned  to 
the  right  or  to  the  left.  If  there  is  only  one  row  of  num- 
bers running  from  o°  to  360°,  and  if  the  vernier  passes 
the  zero  point  in  turning  to  the  second  station,  360° 
must  be  added  to  the  smaller  reading  before  taking  the 
difference.  If  there  are  two  rows  of  numbers  running 
from  o°  to  360°,  it  is  immaterial  which  way  the  tele- 
scope is  turned,  since  that  graduation  may  be  read  which 
increases  in  the  direction  of  the  motion.  It  is  con- 
venient, though  not  necessary  nor  quite  as  accurate,  to 
set  the  vernier  to  read  zero  at  the  beginning,  instead  of 
reading  it  where  it  may  chance  to  be. 

130.  Angles  Measured  More  Accurately.     It  sometimes 
happens    that  an   engineer    desires    an    angle  with    the 
utmost  accuracy.     There   are  two  methods   of  making 
the    observations    when    extreme   accuracy   is   desired; 
viz.,  by  series  and  by  repetition.     One  or  the  other  of  these 
methods  is  always  used  in  measuring  the  principal  an- 
gles   of  a    geodetic    triangulation.     The  principles  in- 
volved are  useful  in  less  accurate  work. 

131.  By  Series.     Sight  upon  the  first  station,  and  read 
both    verniers    to    eliminate    eccentricity.     Sight   upon 


Il8  THE    TRANSIT.  [CHAP.  VII 

the  next  station  to  the  right,  and  read  as  before.  Con- 
tinue around  the  horizon,  reading  upon  each  station, 
and  close  by  reading  upon  the  first  station  again.  If 
the  last  reading  is  the  same  as  the  first,  it  proves  that 
the  instrument  did  not  slip  or  get  moved.  Reverse  in 
altitude  and  azimuth,  turn  on  the  lower  axis  a  little  to 
eliminate  personal  bias,  and  read  upon  the  first  station. 
Proceed  around  the  horizon  towards  the  left,  reading 
upon  each  station  and  closing  upon  the  first.  The  re- 
versal in  azimuth  and  altitude  eliminates  eccentricity  of 
line  of  sight,  error  of  telescope  slide,  and  inclination  of 
horizontal  axis.  Reversing  the  direction  around  the 
horizon  eliminates  any  twist  of  the  tripod  ;  and  shifting 
the  horizontal  circle  diminishes  the  possibilities  of  ac- 
cidental errors  of  graduation.  The  above  observations 
constitute  one  "set." 

To  secure  greater  accuracy  by  increasing  the  number 
of  observations,  and  also  to  eliminate  periodic  errors  of 
graduation,  shift  the  horizontal  circle  an  aliquot  part 
of  the  distance  between  verniers,  and  take  another  set. 
The  amount  that  the  circle  should  be  shifted  between 
sets  is  equal  to  the  distance  between  verniers  divided 
by  the  number  of  sets  to  be  taken.  For  example,  if  the 
angle  is  to  be  read  three  times,  and  if  there  are  two  op- 
posite verniers,  shift  the  circle  60°. 

The  arithmetical  mean  of  the  observed  values  is  to 
be  considered  the  true  angle. 

132.  By  Repetition.  Sight  upon  the  first  station,  read 
both  verniers  ;  and  with  the  upper  motion  turn  to  the 
next  station,  and  read  as  before.  The  last  reading  is 
only  for  a  check.  With  the  lower  motion  turn  back  to 
the  first  station,  the  reading  remaining  unchanged  ; 
then  unclamp  above,  and  turn  forward  again  to  the 
second  station.  The  angle  will  now  have  been  meas- 
ured a  second  time,  but  on  a  part  of  the  circle  adjoin- 
ing that  on  which  it  was  first  measured,  the  second 


ART.  4]  USING  THE  TRANSIT.  119 

beginning  where  the  first  ended.  This  operation  may 
be  repeated  any  number  of  times,  the  circle  being  read 
after  the  last  sight  upon  the  second  point.  The  differ- 
ence of  the  first  and  last  readings  divided  by  the  num- 
ber of  repetitions  gives  the  angle  more  precisely  than 
would  a  single  observation.  Notice  that  the  vernier 
need  be  read  only  at  the  beginning  and  end,  although 
the  second  reading,  as  above,  is  a  valuable  check  in  de- 
termining how  many  times  360°  should  be  added  to  the 
last  reading  in  case  the  vernier  has  passed  o°.  Next 
reverse  in  altitude  and  azimuth,  and  measure  the  an- 
gle as  before,  beginning  however  at  the  second  station. 

This  method  eliminates  all  errors  of  adjustment,  and 
reduces  the  error  of  observation  by  increasing  the  num* 
ber  of  observations.  Of  course,  the  mean  of  the  ob* 
served  values  is  assumed  to  be  the  true  angle. 

133.  Comparison  of  Methods.  Both  methods  seem  to 
be  about  perfect,  as  far  as  the  elimination  of  errors  of 
adjustment,  of  graduation,  and  of  observation  is  con- 
cerned. The  method  by  series  is  preferred  by  most  ob- 
servers for  triangulation  work.  Its  peculiar  advantages 
can  be  fully  realized  only  with  the  precise  instruments 
used  in  that  kind  of  work.  With  ordinary  engineering 
instruments,  there  is  a  limit  beyond  which  it  is  useless 
to  multiply  observations  by  this  method.  For  exam- 
ple, if  the  instrument  reads  to  minutes  and  the  serial 
readings  agree  to  minutes,  the  multiplication  of  obser- 
vations adds  nothing  to  the  accuracy  of  the  result. 

The  method  by  repetition  was  once  a  great  favorite 
with  the  best  engineers,  especially  the  French,  for  tri- 
angulation work  ;  but  the  improvements  in  the  manu- 
facture of  angle  instruments  has  given  precedence  to 
the  method  by  series  for  the  most  accurate  work.  How- 
ever, the  method  by  repetition  is  peculiarly  suited  to 
the  precise  measurement  of  angles  with  a  coarsely- 
divided  circle,  as,  for  example,  a  common  engineer's 


120  THE    TRANSIT.  [CHAP.  VII 

transit.  The  principle  of  this  method  is  certainly  very 
beautiful,  but  its  accuracy  is  largely  dependent  upon 
the  freedom  with  which  the  instrument  turns  on  its 
centers,  and  upon  the  stability  of  the  clamp  and  tangent 
screw.  Under  ordinary  conditions,  the  limit  of  this 
method  is  reached  after  a  few  repetitions. 

134.  TRANSIT   SURVEYING.     Under   this   head  will  be 
discussed    methods    of    doing    transit    work  which    are 
more  or  less  applicable  in  running  a  line  survey — as  for 
a  railroad, — or  in  finding  areas,  or  in  topographical  sur- 
veying.    There  is  no  generally  accepted  method  of  do- 
ing transit  work.     The  three  methods  in  more  or  less 
general  use  will  now  be  considered.     The  first,  for  want 
of  a  better  name,  will  be  called  the  angle  method j  the 
second  may  be  appropriately  named  the  quadrant  method; 
and    the    third    has   been    called  traversing.      The   last 
might  well  be  named  the  full-circle  system. 

135.  Angle  Method.     This  method  consists  in  measur- 
ing and  recording  the  angle  which  each  line  makes  with 
the  preceding  one.     The  angle  measured  may  be  the 
one  included   between  the  two  lines,  or  the  angle  be- 
tween the  second  line  and  the  first  produced.     In  the 
latter    case,    the    telescope    is    sighted    along   the   first 
course,  the  vernier   read,  and   the   telescope    transited.. 
The  telescope  then  points  in  the  direction  of  the  first 
line  produced.     It   is   next   turned  .to   the  second  line, 
and  the  vernier  read.     The  difference  of  the  readings, 
/>.,  the  angle  swept  over,  is  equal  to  the  angle  between 
the  first  course  produced  and  the  second.     This  angle 
is  sometimes  called  the  angle  of  deflection,  and  is  re- 
corded as  a  deflection  to  the  right  or  to  the  left. 

136.  Quadrant   Method.     The   distinguishing   charac- 
teristic   of    this    method    is    that    the    angles    are    read 
and    recorded    as    bearings,    just    as    in    compass    sur- 
veying.      The    meridian    through    the    first    station    is 
obtained  by  reading  the   needle,  or  by  sighting  upon 


ART.  4]  USING    THE    TRANSIT.  121 

some  line  whose  bearing  is  known,  or  it  may  be  as- 
sumed— since  generally  the  object  of  such  surveys  is  not 
so  much  to  get  the  true  bearings  as  to  get  the  relative 
bearings  accurately.  The  bearing  of  any  succeeding 
line  is  found  by  measuring  the  angle  which  it  makes 
with  the  preceding  line  produced,  and  adding  it  to  or 
subtracting  it  from  the  bearing  of  the  preceding  line. 
This  method  is  doubtless  a  survival  from  the  time  when 
the  magnetic  compass  was  the  instrument  ordinarily 
used  in  measuring  horizontal  angles.  The  o°-to-9o° 
numbering  on  the  horizontal  circle  of  transits  is  made 
especially  for  this  system. 

137.  Traversing.  Traverse  surveying,  or  running  a 
traverse,  or  simply  traversing,  is  conducting  a  survey  in 
such  a  way  that  the  readings  of  the  plate  will  show  the 
angles  which  each  line  of  the  survey  makes  with  any 
chosen  reference  line. 

To  run  a  traverse,  set  the  instrument  up  over  the  first 
station — the  end  of  the  first  course.  For  the  present 
we  will  assume  that  the  first  course  is  the  reference 
line,  i.e.,  the  meridian  of  the  survey.  Set  vernier  A  at 
o°,  clamp  the  upper  motion,  turn  the  telescope  upside 
down,  and  with  the  lower  motion  direct  the  line  of 
sight  to  the  other  end  of  the  first  course,  then  clamp 
the  instrument,  and  transit  the  telescope.  Now  loosen 
the  upper  motion,  sight  upon  the  next  station,  and  read 
vernier  A  again.  Move  to  the  next  station,  loosen  the 
lower  motion  (to  prevent  the  possibility  of  disturbing 
the  reading  in  leveling  up),  level  up,  invert  the  tele- 
scope, and  glance  at  the  reading  to  see  that  it  has  not 
changed.  Sight  upon  the  last  station  (using  the  lower 
motion),  and  clamp.  Invert  the  telescope,  loosen  the 
upper  movement,  sight  upon  the  next  station,  and  read. 
Proceed  in  like  manner  for  any  number  of  lines.  Each 
reading  is  the  angle  between  its  corresponding  course 
and  the  first  one. 


122 


THE    TRANSIT. 


[CHAP,  vii 


Fig.  30  shows  the  positions  of  the  several  lines  of  a 
A 


\  L 


FIG.  30. 

traverse  survey,  and  the  following  table  shows  the  notes 
for  the  same. 


Stations. 

Back-Sights. 

Fore-Sights. 

Remarks. 

A 

o°  oo' 

35°  52' 

Azimuths  read  from 

B 

35°  52' 

340°  31' 

vernier     A,     and 

C 

340°  31' 

41°  08' 

counted   from  the 

D 

41°  08' 

270°  oo' 

south  toward   the 

E 

270°  oo' 

180°  oo' 

west. 

The  instrument  is  first  set  at  A,  the  line  O  A  being 
regarded  as  the  first  course.  The  vernier  is  set  at 
zero,  and  a  back-sight  taken  to  O,  which  is  recorded 
opposite  A  in  the  back-sight  column.  The  telescope 
is  then  transited,  the  upper  motion  loosened,  the  tele- 
scope directed  to  B,  and  the  reading  of  this  line,  35°  52', 
recorded  opposite  A  in  the  fore-sight  column  of  the 
table.  After  the  instrument  is  removed  to  B  and  the 
back-sight  taken  upon  A,  the  vernier  is  to  be  read  to 
make  sure  that  it  has  not  been  changed,  and  this 
reading  is  recorded  in  the  back-sight  column.  Writing 
this  down  will  be  evidence  that  the  reading  of  the 
vernier  was  checked;  and  in  actual  work  this  wilt  bo 
found  to  be  an  important  check  against  such  errors  as 
turning  the  wrong  tangent  screw  or  reading  the  wrong 
vernier.  The  angles  marked  in  the  diagram  are  the 


ART.  4] 


USING    THE    TRANSIT. 


123 


corresponding  angles  of  the  fore-sight  column  in  the 
table. 

138.  Instead  of  making  the  first  course  the  reference 
line,  as  above,  any  other  line,  the  meridian  for  exam- 
ple, may  be  taken  as  the  reference  line.  In  this  case 

to 


FIG.  31. 

the  instrument  is  first  set  up  at  the  point  of  beginning, 
the  vernier  set  as  zero,  and  the  first  back-sight  made  in 
the  direction  of  the  meridian.  The  remainder  of  the 
field  work  and  the  record  is  as  before. 

It  makes  but  little  difference  whether  the  first  back- 
sight is  made  towards  the  north  or  the  south  end  of  the 
meridian,  but  a  note  in  the  remarks  column  should  show 
which  way  it  is  made.  However,  it  is  more  common, 
and  therefore  better,  to  count  azimuths  from  the  south 
toward  the  west. 

139.  Comparison  of  Methods.  In  the  time  required  for 
the  field  work,  there  is  very  little  difference  between  the 
three  methods,  the  second  and  third  requiring  slightly 
less  than  the  first. 

The  quadrant  system  is  the  oldest  and  the  one  most 
frequently  used.  Its  chief  advantage  is  in  the  facility 
of  checking  the  vernier  by  the  compass  needle.  One  of 
the  chief  objections  to  this  system  is  that  four  different 
directions  are  designated  numerically  by  the  same  angle. 


124  THE    TRANSIT.  [CHAP.  VII 

The  principal  disadvantage  of  the  full-circle  system 
is  the  greater  labor  required  in  checking  the  vernier  by 
the  compass  needle;  but  even  this  could  be  removed  by 
having  the  compass  graduation  of  the  transit  run  from 
o°  to  360°.  However,  this  is  more  than  overbalanced 
by  the  greater  facility  in  computing  and  platting,  and 
in  the  freedom  from  ambiguity  and  error.  This  system 
is  very  convenient  in  railroad  surveying,  and  in  surveys 
to  find  the  area.*  The  full-circle  system  is  indispensa- 
ble in  topographical  surveying.  In  fact,  the  latter  is 
the  only  kind  of  surveying  in  which  it  is  employed  to 
any  considerable  extent;  but  as  its  advantages  become 
better  understood,  it  will  be  more  fully  adopted. 

140.  SOURCES  OF  ERROR.t     The  errors  of  transit  work 
may  be  classified  as  errors  (i)  of  position,  (2)  of  sighting, 
(3)  of  manipulation,  (4)  of  adjustment,  and  (5)  of  reading. 

141.  Errors  of  Position.     The  instrument  may  not  be 
set  up  over  the  point  about  which  the  angle  is  desired, 
nor  over  the  point  previously  sighted  at.     An  error  of 
an  inch  can  not  make  an  error  of  more  than  3  seconds 
if  the   object   sighted   at   is  a   mile   away  ;  but   it   may 
make  an  error  of  3  minutes  if  the  object  is  only  100  feet 
away.     The   position  of  the  instrument  with  reference 
to  the  true  point  and  the  object  sighted  at  affects  the 
values  of  the  resulting  error. 

142.  Errors  of  Sighting.     The  flag-pole    may  not  be 
vertical,  therefore  sight  as  low  upon  it  as  possible.     The 
intersection   of  the  cross  hairs  may  not  exactly   cover 
the  point,  owing  to  lack  of  care  or  to  parallax  in  the 
instrument  ;  but  in  either  case  the  remedy  is  obvious. 
Always  bring  the  intersection  of  the  cross  hairs  to  bear 
upon  the  point  sighted  at,  for  the  vertical  hair  may  not 
be  vertical. 


*  See  Appendix  II. 

t  For  a  discussion  of  Compensating  vs.  Cumulative  Errors,  see  §  18, 


ART.  4]  USING    THE    TRANSIT.  125 

143.  Errors    of    Manipulation.      The    wrong    tangent 
screw  may  be  turned.     This  is  a  fruitful  source  of  error, 
and  one  difficult  to  discover  and   impossible  to  correct. 
If  the  tangent  screw  has  back-lash,  the  instrument  must 
be    handled    so  as  to   prevent  it.     Error   is   sometimes 
produced  by  the  instrument's  turning  on   the  ball-and- 
socket  joint,  owing  to  the  foot  screws'  not  being  screwed 
up  tight   enough.     The  upper  part  of   the   instrument 
should  move  so  freely  as  not  to  twist  the  tripod. 

144.  Errors  of  Adjustment.     The  points  to  be  consid- 
ered are  eccentricity,  equality  of  standards,  straightness 
of  telescope  slide,  and  adjustment  of  cross  hairs.     The 
eccentricity  is  always   small,  and  easily  eliminated  by 
reading    two  verniers.     The  equality  of  the  standards 
can  affect  only  the  horizontal  angles  between  high  and 
low  points,  and  in  ordinary  practice  the  resulting  error 
will  be  inappreciable.     The  straightness  of  the  slide  and 
the  adjustment  of  the   cross  hairs  are  closely  related  ; 
but  neither  will  produce  error  either  in  measuring  hori- 
zontal angles  between  points  equal  distances  from,  and 
equal  distances  above  or  below,  the  instrument,  or  in 
measuring  vertical  angles  between  points  equally  dis- 
tant from  the  instrument.     An  error  of  adjustment  of 
the  line  of  collimation  produces  an  error  of  double  the 
amount  in  traversing,  or  in  prolonging  a  line  by  back- 
sights  and   fore-sights.     All  errors   of  adjustment  can 
be  wholly  eliminated    by  reversing,  making    a   second 
observation,  and  taking  the  mean. 

145.  Errors  of  Reading.     Errors  may  be  produced  by 
reading  the  wrong  vernier,  the  wrong  end  of  a  double 
vernier,  the  wrong  row  of  numbers,  or  by  reading  28* 
instead  of  32°,  etc.,  or  by  forgetting  to  add  the  half- 
degree  of  the  limb  to  the  reading  of  the  vernier  and 
recording  20'  instead  of  50'. 

146.  LIMITS  OF  PRECISION.    It  is  a  little  difficult  to 
get  sufficient  data  for  a  satisfactory  discussion  of  this 


126  THE    TRANSIT.  [CHAP.  VII 

subject  without  making  observations  specially  for  this 
purpose,  which  is  not  desirable.  Work  is  done  under 
very  different  conditions  as  to  weather,  speed,  instru- 
ments, etc.,  and  results  without  full  information  on  all 
points  are  not  very  valuable. 

The  author's  students,  in  the  prosecution  of  the  ordi- 
nary class  work  in  topographical  surveying,  measured 
the  angles  of  ten  triangles.  The  length  of  the  sides 
varied  between  400  and  1,200  feet.  The  conditions  as 
to  time,  targets,  etc.,  were  about  those  of  actual  prac- 
tice. The  instrument  read  to  minutes,  and  certainly  was 
not  the  best  qualit)^.  All  of  the  errors  enumerated  in 
§§  140-45  were  involved.  This  work  gave  32  seconds 
for  the  probable  error  of  a  single  direction  ;  50  seconds 
for  the  probable  error  of  an  angle  ;  and  86  seconds 
for  the  probable  error  of  the  sum  of  the  three  angles 
of  a  triangle.  The  maximum  error  in  the  closing  of  a 
triangle  was  3^  minutes. 

The  same  class  in  measuring  the  four  angles  around 
a  point  and  the  three  angles  of  a  triangle,  using  chain- 
ing pins  for  targets,  with  sights  about  100  feet  long, 
obtained  results  about  half  as  large  as  those  above. 
The  results  of  traversing,  with  flag-poles  for  targets 
and  sights  varying  between  200  feet  and  800  feet,  gave 
results  a  little  greater  than  half  those  of  the  preceding 
paragraph. 

Ten  measurements  of  an  angle  with  an  engineer's 
transit  reading  to  minutes  (the  reading  was  estimated 
to  30  seconds),  and  "  distinct  signals  at  about  400  feet," 
gave  a  probable  error  of  19  seconds  for  a  single  value 
of  the  angle.  Under  the  same  conditions,  the  probable 
error  of  an  angle  as  measured  with  a  transit  reading 
to  thirds  of  minutes  was  12  seconds.* 

In  locating  the  piers  of  the  bridge  across  the  Ohio 

*  Prof.  Mansfield  Merriman,  in  Engineering  News^  Vol.  10,  p.  621. 


ART.  4]  USING    THE    TRANSIT.  127 

River,  at  Cairo,  111.,  by  triangulation  each  angle  was 
measured  fifteen  times  with  an  engineer's  transif  read- 
ing to  10  seconds,  the  maximum  sight  being  about 
5,000  feet  and  the  average  about  2,500  feet.  The  aver- 
age error  of  closure  of  the  triangles  was  1.5  seconds.* 

In  the  topographical  survey  of  St.  Louis,  Mo.,  the 
angles  were  measured  with  an  engineer's  transit  reading 
to  10  seconds.  Each  series  of  courses  began  and  ended 
at  triangulation  stations,  which  were  about  a  mile  apart. 
The  azimuth  was  checked  by  comparing  it  with  the 
azimuth  of  the  lines  of  the  triangulation.  The  average 
error  of  closure  of  azimuth  was  about  i'  5"  for  each 
series  of  courses,  or  about  n  seconds  per  line.f 

147.  A  transit  is  sometimes  used  to  determine  areas. 
The  legitimate  errors  in  the  balancing  of  the  latitudes 
and  departures  in  transit  surveying  can  be  discussed  as 
for  compass  surveying  (§  52),  the  formulas  deduced  for 
that  case  being  applicable  to  transit  surveying.     Such 
a  discussion  shows  that  ordinary  work  with  the  transit 
is    proportionally    as    accurate    as    the    best    chaining. 
Therefore,  in  finding  areas  by  the  transit,  as  much  care 
must  be  given  to  the  chaining  as  to  the  measurement 
of  the  angles. 

148.  CARE  OF  THE  TRANSIT.     There  are  a  great  many 
small  screws  about  a  transit,  and  the  general  tendency 
is  to  overstrain  them.     This  is  especially  true  of  the 
cross-hair  screws.     All  straining  of  these  screws  beyond 
that  necessary  to  insure  a  firm  seat  is  more  apt  to  cause 
the    instrument    to    lose    than    retain    the    adjustment. 
Overstraining  the  leveling  screws  bends  the  plates  and 
wears  the  screws  unnecessarily.     A  very  common  fault 
is  applying  too  much  force  in  clamping  the  instrument. 
The  operator  frequently  fails  to  appreciate  the  power  of 

*  Journal  of  the  Associated  Engineering  Societies,  Vol.  9,  p.  292, 
f  The  Technograph,  No.  5,  p.  12, 


128  THE    TRANSIT.  [CHAP.  VII 

a  screw.  Some  instrument  makers  invite  this  abuse  of 
their  instruments  by  making  the  heads  of  the  clamping 
screw  too  large.  To  avoid  overstraining  the  clamp,  it 
is  best  to  ascertain  by  trial  the  minimum  force  required 
to  produce  sufficient  hold  for  the  action  of  the  tangent 
screw,  and  then  try  to  clamp  only  slightly  in  excess  of 
this  amount. 

If  the  leveling  or  tangent  screws  get  to  working  hard, 
take  them  out  and  brush  with  soap  and  water.  The 
nuts  can  be  cleaned  by  screwing  a  thin  piece  of  soft 
wood  through  them.  The  tangent  screw  is  intended 
only  to  complete  the  setting,  and  should  never  be 
used  except  to  give  a  slight  movement.  The  tele- 
scope slide  and  the  centers  should  be  examined  occa- 
sionally; and  if  there  is  any  fretting  or  cutting,  take 
the  piece  out  and  burnish  the  rough  place  with  some 
smooth  hard  tool,  as  the  back  of  the  blade  of  a  pocket- 
knife. 

To  lubricate  bearings  exposed  to  the  air,  first  thor- 
oughly clean,  apply  a  little  watch-oil,  vaseline,  or  ren- 
dered ox-marrow,  and  then  wipe  it  off.  The  value  of 
these  three  lubricants  is  in  the  order  enumerated,  vase- 
line and  ox-marrow  being  too  stiff  in  cold  weather,  and 
the  ox-marrow  also  wearing  out  rapidly.  Finely  pow- 
dered plumbago  is  very  good  for  exposed  bearings.  It 
is  a  wise  precaution  to  carry  in  the  field  a  gossamer 
water-proof  bag  to  throw  over  the  instrument  in  case 
of  a  shower,  or  in  case  the  instrument  must  be  left  stand- 
ing for  any  length  of  time  exposed  to  dust. 

To  preserve  the  outer  appearance  of  an  instrument, 
never  use  anything  in  dusting  it  except  a  fine  camel's- 
hair  brush.  To  remove  dust  spots,  first  use  the  camel's- 
hair  brush,  and  then  rub  with  fine  watch-oil,  and  wipe 
dry.  If  the  oil  is  allowed  to  remain  on,  it  will  catch 
dust  and  dirt,  and  thus  do  more  harm  than  good.  On 
reaching  the  office,  after  using  the  instrument,  dust  it 


ART.  4]  USING    THE    TRANSIT.  1 29 

off  generally  with  a  fine  camel's-hair  brush,  examine  the 
centers  and  all  other  principal  movements  to  see  if  they 
run  perfectly  free  and  easy,  and  oil  them  if  necessary. 

To  remove  dirt  and  oxide  that  may  have  accumulated 
on  the  surface  of  a  silver  graduation,  apply  some  fine 
watch-oil,  and  allow  it  to  remain  for  a  few  hours;  then 
take  a  soft  piece  of  old  linen  and  lightly  rub  until  dry, 
but  without  touching  the  edge  of  the.  graduations. 
If,  after  cleaning,  the  silver  surface  should  show  dark 
and  bright  spots  (which  would  interfere  somewhat 
with  the  accurate  reading  of  the  graduation),  barely 
moisten  the  finger  with  vaseline  and  apply  the  same 
to  the  surface;  then  wipe  the  finger  dry  and  lightly 
rub  it  once  or  twice  around  the  circle,  touching  the 
graduation  as  little  as  possible.  Such  cleaning,  how- 
ever, must  be  resorted  to  only  when  absolutely  neces- 
sary, and  then  only  with  the  greatest  care,  as  it  is  too 
apt  to  spoil  the  sharpness  of  the  lines,  which  will  de- 
crease the  accuracy  of  the  reading. 

It  is  not  desirable  to  take  the  instrument  apart  unnec- 
essarily, for  even  if  the  fittings  are  perfect,  it  requires 
considerable  care  to  put  them  together  properly.  A 
little  dust  or  dirt  in  a  joint  or  bearing,  or  a  screw  left 
loose  or  tightened  too  much,  may  damage  the  instrument 
or  cause  errors  in  its  use.  As  long  as  an  instrument 
works  well  and  the  centers  revolve  freely,  it  is  best  not 
to  disturb  it. 

For  hints  on  the  care  of  the  telescope,  see  page  89. 


CHAPTER  VIII. 
SOLAR   TRANSIT. 

149.  THE  solar  transit  is  simply  an  engineer's  transit 
to  which  is  attached  a  solar  apparatus  (§  55).     There  is 
great  diversity  in  the  form   of  the  solar  apparatus  and 
in  the  method  of  attaching  it  to  the  transit. 

ART.  1.     CONSTRUCTION  OF  THE  SOLAR  TRANSIT. 

150.  SAEGMULLEE'S  SOLAR  TRANSIT.    The  latest  form 
of  solar  transit  is  shown  in  Fig.  32.*     It  consists  sim- 
ply of  a  telescope  and  level  attached  to  an  ordinary 
transit  in  such  a  manner  as  to  be  free  to  revolve  in  two 
directions  at  right  angles   to  each  other.      When  the 
transit  telescope  is  horizontal,  the  auxiliary  telescope 
and  its  level  revolve  in  horizontal  and  vertical  planes. 

If  the  transit  telescope  be  brought  into  the  meridian 
and  the  vertical  circle  of  the  transit  be  set  at  the  co- 
latitude,  the  polar  axis  of  the  solar  telescope  will  then 
be  parallel  to  the  axis  of  the  earth  ;  then  if  the  solar 
telescope  be  turned  on  the  polar  axis,  the  solar  sight- 
line  will  describe  a  line  parallel  to  the  equator.  If  the 
solar  sight-line  is  perpendicular  to  the  polar  axis,  the 
line  of  sight  will  describe  the  equator  when  the  solar 

*  Invented  by  G.  N.  Saegmuller,  in  1881.  This  form  is  made  by  G.  N. 
Saegmuller,  Washington,  D.  C.,  and  by  Keuffel  &  Esser  Co.,  New  York 
City ;  and  somewhat  the  same  form  is  made  by  Buff  &  Berger,  Boston, 
Mass. 

130 


ART.   l]     CONSTRUCTION    OF    THE    SOLAR    TRANSIT.  131 


FIG.  32.—  SAEGMUI.LER'S 


ATTACHMENT. 


132  SOLAR    TRANSIT.  [CHAP.  VIII 

telescope  is  revolved  on  the  polar  axis.  If  the  solar 
sight-line  makes  an  angle  with  the  perpendicular  to  the 
polar  axis  equal  to  the  declination  of  the  sun,  then 
when  the  solar  telescope  is  revolved  the  line  of  sight, 
would  follow  the  sun's  path  in  the  heavens  for  the  given 
day,  provided  the  sun  did  not  change  its  declination. 
Therefore,  if  the  solar  telescope  is  set  at  an  angle  with 
the  perpendicular  to  the  polar  axis  equal  to  the  dec- 
lination of  the  sun,  and  the  solar  telescope  is  directed 
to  the  sun,  then  the  terrestrial  sight-line  will  indicate  a 
true  meridian. 

151.  With  the  form  of  solar  apparatus  shown  in  Fig. 
32  it  is  difficult  to  sight  the  solar  telescope  upon  the  sun 
accurately,  owing  to  there  being  no  clamp  and  slow- 
motion  screw  for  the  polar  axis.  The  latest  form  of 
solar  attachment  has  a  clamp  and  tangent  screw  for  the 
polar  axis.  See  Fig.  33. 


FIG.  33. — SAEGMULLER'S  IMPROVED  SOLAR  ATTACHMENT. 

Two  pointers  are  attached  to  the  solar  telescope  to 
aid   in  directing  it  to  the  sun.     They  are  so  adjusted 


ART.   l]     CONSTRUCTION    OF    THE    SOLAR    TRANSIT.  133 

that  when  the  shadow  of  the  one  is  thrown  on  the  other, 
the  sun  will  appear  in  the  field  of  view. 

The  solar  telescope  is  provided  with  colored  glass 
shades  to  protect  the  eye  when  observing.  The  objec- 
tive and  the  cross  hairs  are  focused  in  the  usual  way. 
There  is  a  pair  of  horizontal  cross  hairs  and  also  a  pair 
of  vertical  ones,  between  which  the  image  of  the  sun 
is  brought. 

152.  OTHER  FORMS.  Many  solar  transits  consist  of 
an  ordinary  engineer's  transit  to  which  is  attached  the 
solar  apparatus  of  the  solar  compass  (§  56).  The  solar 
attachment  is  fastened  to  the  center  of  the  horizontal 
axis  of  the  telescope,  or  to  the  end  of  the  horizontal 
axis,  or  below  the  horizontal  plate.  All  of  these  forms 
are  inferior  to  the  solar  transit  shown  in  Fig.  32  for  one 
or  more  of  the  following  reasons: 

i.  Most  of  them  are  more  complicated.  2.  All  of 
them  are  more  difficult  to  adjust.  3.  All  are  deficient 
in  precision.  The  solar  sights  of  a  solar  compass  con- 
sist merely  of  a  small  lens  and  a  piece  of  silver  with 
lines  ruled  on  it  placed  at  the  focus  of  the  lens,  the 
coincidence  of  the  sun's  image  with  the  lines  being 
determined  by  the  unaided  eye-  or,  at  best,  with  a 
simple  magnifying  lens.  This  primitive  telescope  is 
probably  about  as  accurate  as  the  terrestrial  sights  of 
the  common  compass,  but  is  much  less  accurate  than 
the  terrestrial  telescope  of  the  solar  transit.  It  is  obvious 
that  the  substitution  of  a  telescope  for  the  lens  and  the 
lines  of  the  solar  compass  greatly  increases  the  precision 
attainable.  Obviously  the  power  of  the  solar  telescope 
should  be  in  keeping  with  that  of  the  terrestrial  telescope. 
4.  The  solar  telescope  can  be  used  when  the  sun  is 
partly  obscured  by  clouds,  at  which  time  the  ordinary 
solar  apparatus  fails  altogether. 


134  SOLAR    TRANSIT.  [CHAP.  VIII 


ART.  2.     ADJUSTMENTS  OF  THE  SOLAR  TRANSIT.* 

153.  The  adjustments  of  the  solar  transit  consist  of 
two  distinct  operations — the  adjustment  of  the  transit 
and  that  of  the  solar  apparatus. 

The  transit  should  be  in  perfect  adjustment,  particu- 
larly the  plate  levels  (§§  38  and  39),  the  standards 
(§  125),  and  the  zero  of  the  vertical  circle  (§  127). 

154.  ADJUSTMENT  OF  THE  POLAK  Axis.    The  polar  axis 
should  be  vertical  when  the  line  of  collimation  and  the 
horizontal  axis  of  the  transit  are  horizontal.     To  make 
this  adjustment  set  the  vernier  of  the  vertical  circle  to 
read  zero,  and  level  the  instrument   by  means   of   the 
plate  levels.     If  there  is  a  level  under  the  telescope,  the 
verticality  of  the  vertical  axis  can  be  tested  by  revolving 
the  transit  on  its  vertical  axis,  and  noticing  whether  the 
bubble  of  the  level  under  the  telescope  remains  station- 
ary during  the  entire  revolution.     If  it  does  not  remain 
stationary,  correct  half    the  error  by  turning   the  foot 
screws.     Since  the  level  under  the  telescope  is  usually 
more  sensitive  than  the  plate  levels,  it  is  better  to  level 
the  instrument  by  the  latter  than  by  the  former.    When 
these  steps  have  been  correctly  made,  the  plane  of  the 
horizontal   axis  and   the  line  of   collimation   are   hori- 
zontal. 

Then,  to  bring  the  polar  axis  vertical,  revolve  the 
solar  telescope  about  the  polar  axis  and  notice  if  the 
bubble  on  the  small  telescope  maintains  a  constant 
position.  If  it  does  not,  correct  half  the  movement  by 
revolving  the  solar  telescope  on  its  horizontal  axis,  and 
the  other  half  by  means  of  the  adjusting  screws  at 
the  base  of  the  solar  apparatus  (see  Fig.  33,  page  132). 


*  For  general  remarks  upon  adjustments,  see  §  37. 


ART.  3]  USING    THE    SOLAR    TRANSIT.  135 

Notice  that  these  adjusting  screws  are  analogous  to  the 
foot  screws  of  the  main  instrument. 

155.  ADJUSTMENT  OF  THE  CROSS  HAIRS.  The  line  of 
collimation  of  the  solar  telescope  and  the  axis  of  its 
level  should  be  parallel.  Bring  the  terrestrial  telescope 
horizontal  by  setting  the  vernier  of  the  vertical  circle, 
or  by  reading  the  bubble  under  the  telescope.  Place 
the  solar  telescope  as  nearly  as  possible  in  the  plane  of 
the  terrestrial  telescope,  and  make  the  former  horizon- 
tal by  means  of  its  bubble,  and  clamp  it.  Measure  the 
distance  between  the  axes  of  the  two  telescopes,  and 
draw  at  this  distance  from  each  other  two  heavy  black 
lines  on  a  piece  of  paper.  Set  this  piece  of  paper  up, 
with  the  lines  horizontal,  at  a  convenient  distance  from 
the  instrument  and  on  about  the  same  level  as  the 
telescope.  Bring  the  terrestrial  line  of  sight  upon  the 
lower  mark,  and  see  if  the  solar  line  of  sight  falls  upon 
the  upper  mark.  If  it  does  not,  move  the  cross  hairs  of 
the  solar  telescope  until  it  does.  Since  moving  the 
cross  hairs  is  very  liable  to  revolve  the  solar  telescope 
on  its  horizontal  axis,  test  the  adjustment  by  revolving 
back  to  the  horizontal  position,  and  see  If  both  bubbles 
come  to  the  middle  simultaneously. 


ART.  3.     USING  THE  SOLAR  TRANSIT. 

156.  To  DETERMINE  A  MERIDIAN.  Notice  that  an 
observation  with  the  solar  transit  involves  four  quan- 
tities, as  follows:  (i)  the  time  of  day,  i.e.,  the  hour  angle 
of  the  sun;  (2)  the  declination  of  the  sun;  (3)  the  lati- 
tude of  the  place  of  observation;  and  (4)  the  direction 
of  the  meridian.  In  a  general  way,  if  any  three  of  these 
are  known,  the  fourth  may  be  found  by  observation. 
The  prime  object  of  the  solar  transit  is  to  find  the  true 
meridian  when  the  other  three  elements  are  known. 


136  SOLAR    TRANSIT.  [CHAP.  VIII 

157.  The  Time.     The  declination  of  the  sun  is  tabu- 
lated for  Greenwich  or  Washington  noon,  and  hence  a 
correction  must  be  applied  to  find  the  declination  at  the 
time  of  the  observation.     An  error  of  30  minutes  in  the 
time  can  not  make  an  error  of  more  than  30"  in  the 
declination,  and  the  average  error  will  be  about  15", 
which  is  less  than  can  be  laid  off  on  the   instrument. 
Therefore  it  is  sufficient  if  the  error  in  the  time  does 
not  exceed  30  minutes;  and  ordinarily  there  will  be  no 
difficulty  in  finding  it  much  closer. 

158.  The  Decimation.     The  declination  of  the  sun,  and 
also  the  hourly  change  in  the  declination,  is  given  for 
each  day  of  the  year,  in  the  "American  Ephemeris  and 
Nautical  Almanac,"*  for  both  Greenwich  and  Washing- 
ton  mean    noon.       Hence,   to    find    the    declination    at 
any  given   time,  it    is    necessary  to    know  the   time  of 
day  and  the  longitude  from  either  Greenwich  or  Wash- 
ington; and  since  the  time  generally  employed  in  this 
country,    *>.,    standard    time,    differs    from    Greenwich 
time  by   an  integral   number  of    hours,  it   is  better  to 
use  the  Greenwich  declination.     Therefore,  for  a  point 
on,  say,  the  meridian  of  New  Orleans   (90°  west  from 
Greenwich),  the  declination  given  in  the  ephemeris  for 
Greenwich    noon    is    the    declination    at  6  a.m.  at  the 
place  of   observation.     For  a  point  either  side  of  the 
90th  meridian,    the    standard    time   does    not    indicate 
the  true  hour  angle  of  the  sun,  i.e.,  there  is  a  difference 
between  standard  and  local  time;  and  hence  for  a  place 
west  of  the  90th  meridian    the  declination  at  noon  at 
Greenwich  is  the  declination  at  some  time  before  6  a.m., 
and  for  a  place  east  of  the  90th  meridian   the  declina- 
tion at  Greenwich  noon  is  the  declination  at  some  time 
after  6  a.m. 


*  Issued  several  years  in  advance  by  the  U.  S.  Government,  and  for  sale  by 
book  dealers.     Price  $1.25. 


ART.  3]  USING    THE    SOLAR    TRANSIT.  137 

Since  the  difference  between  standard  and  local  time 
is  ordinarily  less  than  30  minutes,  and  since  the  hourly 
change  in  declination  is  less  than  60  seconds  of  arc,  it 
is  sufficiently  exact  to  assume  that  the  declination  of 
the  sun  given  in  the  ephemeris  for  Greenwich  noon  for 
any  day  is  the  declination  at  7,  6,  5,  or  4  a.m.  of  the 
same  date  according  as  the  point  is  situated  in  the  Eas- 
tern, Central,  Mountain,  or  Western  time  belt.  Thus  if 
the  point  of  observation  is  in  the  Central  time  belt,  the 
declination  given  in  the  ephemeris  may  be  assumed  to  be 
the  declination  at  6  a.m.  at  the  place  of  observation. 
Of  course,  this  is  strictly  true  only  for  points  on  the 
governing  meridian  of  each  time  belt. 

Knowing  the  declination  for  a  given  time  of  day  at 
the  point  of  observation,  to  find  the  declination  for  any 
other  time  of  day  it  is  only  necessary  to  multiply  the 
hourly  change  by  the  difference  between  the  time  for 
which  the  declination  is  given  and  the  time  for  which 
it  is  required,  and  add  it  (algebraically)  to  the  tabu- 
lated value.  For  example,  if  the  declination  at  Green- 
wich noon  on  November  19,  is  —  19°  28'  24",  and  the 
hourly  change  is  —  35",  what  is  the  declination  at 
ii  a.m.  at  Chicago?  The  declination  at  Greenwich 
noon  corresponds  (nearly)  to  the  declination  at  6  a.m.  at 
Chicago;  and  hence  a  change  for  5  hours  (from  6  a.m. 
to  ii  a.m.)  must  be  allowed  for.  Therefore  the  required 
declination  is  —  19°  28'  24"  plus  (—  35"  X  5),  which 
equals  —  19°  31'  19".  In  a  similar  way  the  declination 
for  any  hour  may  be  found. 

159.  To  Correct  the  Declination  for  Refraction.  Owing 
to  refraction,  all  celestial  objects  appear  higher  than 
they  really  are.  The  declination  to  be  set  off  on  the 
solar  apparatuses  the  apparent  and  not  the  real  decli- 
nation, and  hence  the  declination  as  found  above,  before 
being  set  off  on  the  instrument,  must  be  corrected  for 
refraction. 


138  SOLAR    TRANSIT.  [CHAP.  VIII 

The  refraction  is  zero  for  a  point  in  the  zenith,  about 
i'  for  an  altitude  of  45°,  and  about  34"  at  the  horizon. 
It  varies  greatly  with  the  temperature,  pressure,  and 
hygrometrical  condition  of  the  atmosphere, — particu- 
larly near  the  horizon.  It  is  for  this  reason  that  all 
astronomical  observations  made  near  the  horizon  are 
very  uncertain.  Tables  of  mean  refraction  are  fre- 
quently given  in  text-books  on  astronomy,*  in  loga- 
rithmic tables,  etc. 

Notice  that  refraction  changes  the  apparent  altitude, 
which  is  measured  perpendicular  to  the  horizon,  while 
we  desire  to  know  the  effect  upon  the  apparent  decli- 
nation, which  is  measured  perpendicular  to  the  equa- 
tor of  the  heavens.  When  the  sun  is  on  the  meridian, 
the  change  in  altitude  has  its  full  effect  in  changing  the 
declination,  but  at  other  times  the  change  in  declination 
is  less  than  the  change  in  altitude.  The  correction  to 
the  declination  due  to  refraction  is 

C"  =  57"  cot  (d  +  N),\    .    .    ,    .    (i) 

where  d  •=  declination, — plus    when    north    and   minus 
when  south.     N  is  an  auxiliary  angle  such  that 

tan  -A7"  =  cot  0  cos  /,      ....     (2) 

where  0  is  the  latitude,  and  /  is  the  hour  angle. 

160.  By  means  of  equations  (i)  and  (2)  the  refraction 
correction  can  be  computed  for  any  latitude,  hour  angle, 
and  declination.  Messrs.  W.  and  L.  E.  Gurley,  Troy, 
N.  Y.,  publish  a  table  giving  this  correction  for  each 
2-J°  of  latitude  from  30°  to  57^°,  for  each  5°  of  declina- 
tion, and  for  integral  hour  angles  from  o  to  5. 

*  For  example,  Loomis's  Practical  Astronomy,  p.  364,  and  Chauvenet's 
Spherical  and  Practical  Astronomy,  vol.  ii,  pp.  604-7. 

t  See  Chauvenet's  Spherical  and  Practical  Astronomy,  vol.  i,  p.  171. 


ART.  3]  USING    THE    SOLAR    TRANSIT.  1^9 

G.  N.  Saegmuller,  Washington,  D.  C.,  publishes  an- 
nually for  gratuitous  distribution  a  small  pamphlet  giv- 
ing the  declination  of  the  sun  and  its  hourly  change  for 
each  day  of  the  year.  In  the  margin  of  the  table  is 
given  the  refraction  correction  for  40°  of  latitude  from 
o  hours  to  5  hours  of  hour  angle.  In  a  supplemental 
table  is  given  a  series  of  co-efficients  by  which  the  re- 
fraction correction  for  40°  of  latitude  must  be  multi- 
plied to  obtain  the  refraction  correction  for  any  other 
latitude.  As  very  complete  explanations  accompany 
these  tables,  and  as  it  is  necessary  to  have  a  table  of 
declinations  for  the  current  year,  it  is  not  thought  wise 
to  consider  this  subject  further  here,  but  the  reader 
who  has  any  considerable  solar  work  to  do  is  advised 
to  send  for  this  little  pamphlet,  the  title  of  which  is 
"  Solar  Ephemeris  and  Refraction  Tables." 

161.  The  Latitude.     The  latitude  is  required  that  the 
polar  axis  may  be  set  parallel  to  the  axis  of  the  earth. 
To  find  the  latitude,  a  few  minutes  before  the  meridian 
passage  of  the  sun  level  the  transit  carefully  and  point 
the  telescopes  toward  the  south.     Then  incline  the  ter- 
restrial telescope — if  the  declination  is  north  depress 
it,  and  if  south   elevate   it, — until   the  reading  of  the 
vertical   circle   is  equal  to   the   declination  of  the  sun 
uncorrected  for  refraction ;  and  bring  the  solar  telescope 
into  the  vertical  plane  of  the  terrestrial  telescope,  level 
it  carefully,  and  clamp   it.     By  moving  the  terrestrial 
telescope  in  altitude  and  azimuth,  bring  the  image  of 
the  sun   between  the  hairs  of  the  solar  telescope,  and 
keep  it  there  until  the  sun  ceases   to  rise.     Then  the 
reading  of  the  vertical  circle,  corrected  for  refraction, 
is  the  co-latitude. 

162.  The  Observation.     First  focus  the  solar  telescope 
for  distinct  vision   of  the   cross  hairs,  and    then   care- 
fully focus  it    upon  the  sun.     It  is  necessary  to  make 
these  adjustments   before  setting  off    the  declination, 


140  .  SOLAR    TRANSIT.  [CHAP.   VIII 

since  the  solar  telescope  is  liable  to  be  displaced  if  they 
are  made  afterwards. 

Set  off  the  declination  by  inclining  the  transit  tele- 
scope until  the  vertical  circle  reads  the  declination — if 
the  sun-  is  south  of  the  equator  elevate  the  transit  tele- 
scope, and  if  north  depress  it.  Without  disturbing  the 
position  of  the  transit  telescope,  bring  the  solar  tele- 
scope into  the  vertical  plane  of  the  transit  telescope, 
and  also  to  a  horizontal  position  by  means  of  the  level 
on  the  solar  telescope.  The  two  telescopes  will  then 
make  an  angle  with  each  other  equal  to  the  declination, 
and  the  inclination  of  the  solar  telescope  to  its  polar 
axis  will  be  equal  to  the  polar  distance  of  the  sun. 
Without  disturbing  the  relative  position  of  the  two 
telescopes,  set  the  vernier  of  the  vertical  circle  to  the 
co-latitude  of  the  place  of  observation. 

By  revolving  the  transit  on  its  vertical  axis  and  the 
solar  apparatus  about  its  polar  axis,  taking  great  care 
not  to  revolve  either  telescope  on  its  horizontal  axis, 
bring  the  image  of  the  sun  into  the  solar  telescope,  and 
then  the  transit  telescope  must  be  on  the  meridian.  If 
the  vertical  motion  of  the  transit  be  clamped,  the  tran- 
sit telescope  may  be  turned  down  and  the  meridian 
marked;  or  the  horizontal  circle  may  be  read,  and  the 
angle  which  any  line  makes  with  the  true  meridian  can 
be  found. 

163.  SOURCES   OF   ERROR.*     There    are    three    sources 
of  error  in  addition  to  those  of  adjustment  and  manip- 
ulation of  the  instrument,  viz.:   (i)  error  of  declination, 
(2)  error  of  latitude,  and  (3)  error  of  refraction. 

164.  Declination  and  Latitude.      Table  II f  (page  141) 
shows  the  errors  in  the  direction  of   the   meridian   due 
to  an  error  of   i  minute   in   the  latitude  or  declination. 


*  For  a  discussion  of  Cumulative  vs.  Compensating  Errors,  see  §  18. 
t  From  "  Theory  and  Practice  of  Surveying,"  by  J.  B.  Johnson,  by  per- 
mission. 


ART.  3]  USING    THE    SOLAR    TRANSIT. 


TABLE    II. 

ERRORS  IN  AZIMUTH  DETERMINED  BY  SOLAR  TRANSIT  FOR  i  MINUTE 
ERROR  IN  DECLINATION  OR  LATITUDE. 


HOUR. 

FOR  i  MIN.  ERROR  IN 
DECLINATION. 

FOR  i  MIN.  ERROR  IN 
LATITUDE. 

Lat.  30°. 

Lat.  40°. 

Lat.  50°. 

Lat.  30°. 

Lat.  40°. 

Lat.  50°. 

1.1.30  a.m.  1 
12.30  p.m.  ) 

n  a.m.  .  .  ,  / 
i  p  m     ,    .  C 

Min. 

8.85 

4.46 
2.31 
1.63 
1-34 

1.20 

I.  IS 

Min. 
10.00 

5.05 

2.61 

1.85 

1.51 
1.35 
1.30 

Min. 
12.90 

6.OI 

3-n 

2.  2O 
I.  80 

1.61 

1.56 

Min. 
8.77 

4-33 
2.00 

I-I5 
0.67 

0.31 
0.00 

Min. 
9.92 

4.87 
2.26 
1.30 
0.75 
0-35 

0.00 

Min. 
II.80 

5-80 
2.70 
I.56 
O.gO 

0-37 
0.00 

10  a.m.  .  .  .  ) 

2  n  m   .           C 

Q  a  m   .        ) 

3  p.m.   ...  f 
8  a.m  ) 

4n  m               f 

7  a.m  .      .  ) 

5p.m  f 

6  a.m  ) 
6  p.m  f 

Several  important  conclusions  may  be  drawn  from  this 
table  and  the  equations  from  which  it  was  deduced. 

1.  "  The  solar  apparatus  should  never  be  used  between 
ii  a.m.  and  i   p.m.,  and  preferably  not  between  10  a.m. 
and  2  p.m.,  if  the  best  results  are  desired. 

2.  "  At  6  a.m.  and  6  p.m.,  when  the  line  of  collima- 
tion  lies  in  a  plane  at  right  angles  to  the  plane  of  the 
meridian,  no  small  error  in  the  latitude  will  affect  the 
accuracy  of  the  result. 

3.  "  The  best  times  of  day  for  using  the  solar  apparatus 
are  from  7  to   10  a.m.  and  from  2  to  5  p.m.     So  far  as 
the  instrumental  errors  are  concerned,  the  greater  the 
hour   angle  the  better  the  observation;  but  when  the 


142  SOLAR  TRANSIT.  [CHAP,  vni 

sun  is  near  the  horizon,  the  uncertainties  in  the  refrac- 
tion may  cause  unknown  errors  of  considerable  size. 

4.  "  For  a  given  error  in  the  setting  for  declination 
or  latitude  the  resulting  error  in  azimuth  will  have  op- 
posite signs  in  forenoon  and  afternoon.     If,  therefore,  a 
jo-o'clock  azimuth  is  in  error  5'  in  one  direction  from 
erroneous    settings,  a    2-o'clock    observation    with    the 
same  instrument  should  give  an  azimuth  5'  in  error  in 
the  opposite  direction. 

5.  "  If  the   declination  angle  be  erroneously  set  off, 
and  the  latitude  angle  be  also  affected  by  an  equal  error 
in  the  opposite  direction,  then  the  two  resulting  errors  in 
azimuth  will  nearly  balance  each  other. 

6.  "  If  the  instrument  is  out  of  adjustment,  the  lati- 
tude found  by  a  meridian  observation  will  be  in  error; 
but  if  this  observed  latitude  be  used  in  setting  off  the  co- 
latitude,  the  instrumental  error  is  eliminated.     There- 
fore always  use  for  the  co-latitude  that  given  by  the 
instrument  itself  in  a  meridian  observation." 

165.  Refraction.  The  refraction  correction  to  the  de- 
clination (§  159)  is  computed  on  the  assumption  that 
the  refraction  is  a  mean,  whereas  the  actual  refraction 
at  any  time  and  place  may  differ  considerably  from  the 
mean  or  average.  This  difference  is  liable  to  be  very 
much  greater  at  low  than  at  high  altitudes.  For  this 
reason  no  observations  should  be  made  within  20°  of 
the  horizon,  and  preferably  not  within  30°.  Fortu- 
nately most  solar  work  is  done  in  the  summer,  when  the 
sun  is  high  in  the  heavens,  and  this  limitation  is  less 
serious  then  than  in  the  winter. 

Any  error  in  the  refraction  correction  to  the  declina- 
tion has  the  same  effect  as  an  equal  error  in  the  decli- 
nation. The  refraction  correction  may  be  computed  by 
means  of  equations  (t)  and  (2),  page  138;  and  the  effect 
of  any  assumed  per  cent  of  error  in  this  correction  may 
be  determined  by  an  inspection  of  Table  II,  page  141. 


ART.  3]  USING    THE    SOLAR    TRANSIT.  143 

For  example,  if  the  observation  be  made  in  latitude  40° 
on  September  23  at  7  a.m.,  the  refraction  correction 
will  be  a  little  more  than  3';  and  if  we  assume  that  the 
refraction  may  be  in  error  20  per  cent  (probably  a  fair 
assumption  for  this  altitude),  the  refraction  correction 
may  be  in  error  nearly  40",  and  from  Table  II  we  see 
that  the  corresponding  error  in  azimuth  would  be  a 
trifle  over  50". 

If  the  observations  are  not  made  within  20°,  or, 
better,  30°,  of  the  horizon,  and  if  the  limitations  in 
§  164  are  observed,  the  error  due  to  refraction  can  not 
be  serious. 

166.  LIMITS  OF  PRECISION.     The  author's  students  in 
ordinary  class  work  adjust  their  own  instruments,  and 
determine  a  meridian  with  the  Saegmuller  solar  attach- 
ment (the  latitude  and  time  being  known  accurately) 
with  a  maximum  error  of  4'  to  5'  between  observations 
made  in  quick  succession,  with  an  average  error  of  2' 
to  3'.     Eight   students    made    three  observations  each, 
and  the  greatest  difference  between   the  mean  of  each 
three  and  the  true  meridian  (as  determined  by  obser- 
vations with   an    astronomical    transit)   was    4'. 6.     The 
average  difference  was  i'.6,  the  probable  error  (Appen- 
dix III)  of  the  mean  of  three  observations  was  i'.8,  and 
the  probable  error  of  a  single  observation  was  I'.o. 

167.  The  following  results  by  Professor  J.  B.   John- 
son, of  Washington  University,  St.   Louis,  Mo.,  may  be 
taken  as  the  best  than  can  be  done  :  * 

"  In  order  to  determine  just  what  accuracy  was  possi- 
ble with  a  Saegmuller  solar  attachment,  I  spent  two 
days  in  making  observations  on  a  line  whose  azimuth 
had  been  determined  by  observations  on  two  nights  on 
Polaris  at  elongatioa,  the  instrument  being  reversed  to 
eliminate  errors  of  adjustment.  Forty-five  observations 

*  Journal  of  the  Association  of  Engineering  Societies,  Vol.  5,  p.  35. 


144  SOLAR    TRANSIT.  [CHAP.  VIII 

were  made  with  the  solar  attachment  on  October  24, 
1885,  from  9  to  10  a.m.,  and  from  1.30  to  4  p.m.,  and  on 
November  7  forty-two  observations  between  the  same 
hours. 

"  On  the  first  day's  work  the  latitude  used  was  that 
obtained  by  an  observation  on  the  sun  at  its  meridian 
passage,  being  38°  39',  and  the  mean  azimuth  was 
20"  in  error.  On  the  second  day,  the  instrument  hav- 
ing been  more  carefully  adjusted,  the  latitude  used 
was  38°  37',  which  was  supposed  to  be  about  the  true 
latitude  of  the  point  of  observation.  It  was  afterwards 
found  this  latitude  was  38°  37'  15",  as  referred  to  Wash- 
ington University  Observatory,  so  that  when  the  mean 
azimuth  of  the  line  was  corrected  for  the  15"  error  in 
latitude  it  agreed  exactly  with  the  stellar  azimuth  of 
the  line,  which  might  have  been  10"  or  15"  in  error. 
On  the  first  day  all  the  readings  were  taken  without  a 
reading  glass,  there  being  four  circle  readings  to  each 
result.  On  the  second  day  a  glass  was  used.  On  the 
first  day  the  maximum  error  was  4',  the  average  error 
was  o'.8,  and  the  probable  error  of  a  single  observation 
was  also  o'.8.  On  the  second  day  the  maximum  error 
was  2'.7,  the  average  error  was  i',  and  the  probable 
error  of  a  single  observation  was  o'.86.  The  time  re- 
quired for  a  single  observation  is  from  three  to  five 
minutes." 

168.  THE  SOLAR  TRANSIT  IN  MINE  SURVEYING.  Since 
the  standards  of  the  solar  telescope,  as  ordinarily  made, 
are  long  enough  to  allow  the  small  telescope  to  sight 
past  the  horizontal  plates  of  the  transit,  the  solar  at- 
tachment can  be  used  for  oblique  or  vertical  sighting, 
as  is  frequently  required  in  mine  surveying. 

If  the  solar  transit  is  used  for  this  purpose,  it  should 
be  adjusted  as  described  in  §§  153-155,  and  the  lines  of 
sight  of  the  two  telescopes  should  lie  in  the  same 
vertical  plane.  The  last  adjustment  may  be  made  by 


ART.  3]  USING    THE    SOLAR    TRANSIT.  145 

leveling  the  instrument  and  sighting  both  telescopes 
upon  a  plumb-line.  When  this  adjustment  has  been 
made  the  lines  of  collimation  of  the  two  telescopes  are 
parallel,  and  any  angle,  say  90°  from  the  horizontal, 
may  be  set  off  on  the  vertical  circle,  and  the  sight  made 
through  the  small  telescope.  All  instrumental  errors 
are  eliminated  if,  after  making  one  observation,  the 
transit  is  reversed  on  its  vertical  axis  and  another  ob- 
servation is  made, — the  mean  of  the  two  points  being  the 
correct  one. 


CHAPTER  IX 
PLANE  TABLE. 

ART.  1.     CONSTRUCTION. 

169.  In  its  simplest  form,  the  plane  table  consists  of 
a  drawing  board  mounted  on  a  tripod,  on  which  lines 
are  drawn  to  represent  the  direction  of  any  object  as 
indicated  by  a  ruler  placed  so  as  to  point  to  the  object. 
Any  other  parts  are  mere  additions  to  render  the  oper- 
ations more  convenient  and  precise.* 

170.  COMPLETE  PLANE   TABLE.     Fig.  34  (page  147) 
shows  one  of  the  most  elaborate  forms  of  plane  table. 
An  instrument  of    this  form   is   virtually   a    transit    in 
which  the  horizontal  circle  is  replaced  by  the  drawing 
board,  the  lines  being  drawn  upon  the  paper  instead  of 
being  read  from  the  graduated  limb.     The  lower  plate 
of  the  transit  is  expanded,  and  becomes   the   drawing 
board;    and    the    vernier    or    index    is    replaced    by    a 
straight-edge,   which   with    the    telescope    and  vertical 
arc  is  commonly  called  the  alidade. 

The  alidade  is  usually  about  20  inches  long  and  2-J 
inches  wide.  At  the  center  is  a  standard  which  sup- 
ports the  telescope  and  which  serves  as  a  handle  for 
the  alidade.  In  the  best  plane  tables  the  telescope  is 


*  "  The  invention  of  the  plane  table  is  ascribed  to  Praetorious  in  1537,  but 
the  first  published  description  appears  to  be  that  of  Leonhard  Zubler  in  1625, 
who  ascribes  the  '  beginning '  of  the  instrument  to  one  Eberhart,  a  stone- 
mason." 

146 


ART*   l]  CONSTRUCTION. 


147 


' 


FIG.  34.— COMPLETE  PLANE  TABLE. 


148  PLANE    TABLE.  [CHAP.  IX 

equal  to  that  on  an  ordinary  transit.  In  s,ome  forms 
the  telescope  does  not  transit  on  its  horizontal  axis,  but 
is  reversed  in  azimuth  by  lifting  it  out  of  its  bearings. 
The  telescope  has  no  lateral  movement  with  respect  to 
the  ruler,  but  both  may  be  moved  at  pleasure  on  the 
table.  The  telescope  is  movable  in  a  vertical  plane,  and 
is  provided  with  a  vertical  circle.  In  the  form  shown 
in  Fig.  34,  the  telescope  is  mounted  centrally  over  the 
standard.  In  some  forms  the  line  of  sight  is  placed 
over  the  beveled  edge  of  the  ruler,  though  this  is  not 
essential.  It  is  only  necessary  that  they  should  have  a 
fixed  horizontal  angle  with  each  other. 

The  mechanism  connecting  the  board  with  the  tripod 
is  the  most  important  part  of  the  plane  table.  The 
board  must  be  supported  rigidly  in  a  horizontal  plane, 
and  also  be  free  to  move  in  that  plane.  The  diffi- 
culties of  satisfying  these  conditions  are  increased  by 
the  weight  and  possible  eccentric  position  of  the  ali- 
dade. The  instrument  should  also  have  a  substantial 
clamp  and  tangent  screw  for  the  motion  in  azimuth. 
In  addition,  portability  must  be  considered.  The  form 
shown  in  Fig.  34  secures  great  stability  with  little 
weight,  and  allows  the  facilities  necessary  for  manipu- 
lation. Plane  tables  of  this  class  are  usually  supported 
upon  three  leveling  screws. 

A  declinator,  a  small  box  carrying  a  needle  which 
can  swing  10°  or  15°  either  side  of  the  zero  line,  should 
accompany  the  instrument  for  use  in  determining  the 
magnetic  meridian.  The  zero  line  being  parallel  to  one 
edge  of  the  box,  the  magnetic  meridian  may  at  once  be 
marked  down  on  any  portion  of  the  map,  and  the  bear- 
ing of  any  intersecting  line  may  be  determined  by  a 
protractor.  Sometimes  the  needle  is  capable  of  swing- 
ing through  360°,  in  which  case  the  magnetic  bearing 
of  any  line  may  be  read  with  the  compass-box  alone. 

The  board  varies  greatly  in   size,  but  usually  is  not 


ART.    l]  CONSTRUCTION.  149 

larger  than  24  by  30  inches.  It  is  very  important  that 
the  board  should  be  made  of  well-seasoned  wood  and 
be  so  constructed  as  not  to  warp.  The  securing  of  a 
plane  upon  which  to  work  is  one  of  the  most  difficult 
conditions  to  fulfill  in  the  construction  of  a  plane  table. 
Brass  and  plate  glass  have  been  used  for  this  purpose 
in  Europe,  but  make  the  table  excessively  heavy.  The 
paper  is  fastened  to  the  board  in  any  of  several  ways  : 
(r)  by  thumb-tacks  screwed  into  a  metal  socket  set  in 
countersunk  holes  so  as  to  place  the  head  of  the  tack 
or  screw  out  of  the  way  of  the  ruler  ;  (2)  by  wrapping  it 
around  rollers  at  the  ends  of  the  board  ;  (3)  by  pressing 
the  end  of  the  sheet  against  the  end  of  the  board  with 
a  strip  of  wood  or  metal  connected  to  the  board  by 
screws  ;  (4)  by  movable  spring  clamps  gripping  the  edge 
of  the  board;  (5)  by  pasting  it  down. 

171.  LIGHT  PLANE  TABLE.    The  form  shown  in  Fig. 

34  is  quite  heavy  and  difficult  to  handle  on  rough  or 
obstructed  ground.  The  mechanism  connecting  the 
board  to  the  tripod  is  the  chief  source  of  weight.  To 
reduce  the  weight  of  these  parts,  the  form  shown  in 
Fig-  35  was  invented.*  In  the  figure,  a  represents  a 


FIG.  35. — GURLEY'S  PLANE  TABLE. 

hemispherical  concave  metal  cup  fastened  by  screws  to 
the  wood  top  of  the  tripod,  and  b  a  convex  hemispheri- 
cal cup  fitting  into  the  cup  a  and  being  clamped  to  it  at 

*  Patented  and  manufactured  by  W.  &  L,  E,  Gurley,  Troy,  N,  Y, 


150  PLANE    TABLE.  [CHAP.  IX 

will  by  the  clamping  pieces  and  nut  d.  A  strong  spiral 
spring  in  the  hollow  cylinder  between  c  and  d  holds  the 
two  spherical  surfaces  of  the  two  pieces  together,  and 
also  allows  the  easy  movement  of  the  one  within  the 
other.  The  flange  of  the  cup  b  supports  the  board  and 
is  connected  with  its  under  surface  by  three  segments 
of  brass,  two  of  which  are  shown  at  e,  e.  These  are 
brought  down  firmly  upon  the  shoulder  of  the  flange 
by  capstan-head  screws  as  shown,  or  released  at  will, 
thus  allowing  the  board  to  be  moved  horizontally  when 
desired. 

Fig.  36  shows  an  improvement  of  the  preceding  form, 


FIG.  36.— JOHNSON'S  PLANE  TABLE. 

which  has  been  adopted  by  the  U.  S.  Geological  Sur- 
vey. The  arrangement  is  essentially  the  same  as  in 
Fig.  35,  with  the  addition  of  the  winged  nut  g.  When 
it  is  desired  to  level  the  table,  the  clamping  nut  d  is 
released,  the  board  is  brought  into  position,  and  then 
securely  clamped  by  the  same  nut.  When  it  is  desired 
to  turn  the  table  in  azimuth,  the  nut  g  is  loosened, 
which  leaves  the  hemispherical  surface  b  free  to  move 
around  the  concave  part  a  of  the  tripod  head.  The 
only  objection  to  this  form  of  plane  table  is  the  impos- 
sibility of  leveling  the  board  sufficiently  for  the  accu- 
rate determination  of  vertical  distances  with  the  vertical 
circle  of  the  alidade.  For  work  to  which  this  limita- 
tion does  not  apply,  this  is  a  very  excellent  form  of 
instrument, 


ART.    l] 


CONSTRUCTION. 


172.  HOME-MADE  PLANE  TABLE.  A  very  fair  plane 
table  can  be  made  very  cheaply,  if  the  engineer  has  a 
transit  or  leveling  instrument  in  which  the  upper  part 
is  detachable  from  the  leveling  screws.  This  requires 
a  good  drawing  board,  say  18  by  20  inches,  to  the  under 
side  of  which  is  screwed  a  casting  of  iron  or  brass  simi- 
lar to  Fig.  37.  The  conical  portion,  0,  should  be  made 
to  fit  the  socket  from  which  the  axis  of  the  level  or 


a 


FIG. 


37- 


transit  is  taken  ;  and  the  cylindrical  portion,  c  c,  should 
fit  the  lower  clamp.  The  alidade  may  be  made  of  wood, 
similar  to  that  shown  in  Fig.  38.  One  slit  is  narrow, 


FIG.  38.— ALIDADE. 


the  other  is  wide  and  has  a  thread  through  the  middle. 
The  scale  can  be  obtained  by  attaching  a  printed  paper 


152  PLANE   TABLE.  [CHAP.  IX 

scale.  It  is  convenient  to  have  a  separate  scale  also. 
A  level  vial  may  be  obtained  of  an  instrument  maker  for 
a  few  cents,  and  may  be  fastened  in  the  alidade  with 
plaster  of  Paris.  In  fixing  the  vial  in  position,  remem- 
ber that,  as  usually  made,  level  tubes  are  convex  on  one 
side  and  concave  on  the  opposite.  The  former  should 
be  up,  else  the  bubble  will  run  to  one  end  or  the  other, 
and  never  stop  in  the  middle. 

It  is  sometimes  necessary  to  place  a  point  on  the 
board  exactly  over  the  corresponding  point  on  the 
ground,  which  requires  that  the  plumb-bob  shall  be  sus- 
pended from  the  under  side  of  the  board  immediately 
below  the  point  given  on  the  upper  surface.*  To  aid  in 
doing  this,  a  plumbing  bar,  or  frame,  similar  to  Fig.  39, 


FIG.  39.— PLUMBING  BAR. 

is  very  convenient.  It  is  made  of  two  light  bars  of 
wood  fastened  together  by  a  hinge  at  C.  A  is  a.  piece 
of  metal  sharpened  to  a  vertical  edge.  B  is  a  piece  of 
metal  fastened  into  the  end  of  the  arm  B  C,  and  shaped 
to  carry  a  plumb-line.  The  point  A  is  placed  upon  the 
point  on  the  paper,  and  the  plumb-line  is  suspended  from 
B.  The  plumbing  bar  is  so  made  that  if  the  under  side 
of  the  upper  arm  is  horizontal,  the  line  joining^  and  B  is 
vertical.  The  length  of  the  arm  A  C  should  be  a  little 
greater  than  half  the  greatest  dimension  of  the  board, 
and  A  B  should  be  a  little  greater  than  the  distance  from 
the  top  of  the  board  to  the  bottom  of  the  tripod  head. 

*  For  a  method  of  obviating  this  difficulty,  see  §  188. 


ART.    2J  TESTS    AND    ADJUSTMENTS. 


ART.  2.     TESTS  AND  ADJUSTMENTS.* 

173.  THE  SIGHTS.     The   sights  of  the  elementary  ali- 
dade (§  172)  should  be  perpendicular  to  its  base.     This 
can  be  tested  sufficiently  with  a  try-square.     If  this  con- 
dition is  not  satisfied,  the  slits  will  not  be  vertical  when 
the  board    is  level,  and   sighting  through   the  top  of 
one  and  the  bottom  of  the  other  will  give  a  different 
line  from  that  given  by  sighting   through  the  tops  or 
the  bottoms  of  both. 

174.  EDGE  OF  RULER.     The  edge  of  the  ruler  should 
be  a  straight  line.     To  test  this,  place  the  rule  upon  a 
smooth  surface  and  draw  a  line  along  the  edge  ;  then 
reverse   the  rule  end  for  end,  place  the  edge  upon  the 
line,  and  again  draw  a  line.     If  the  two  lines  coincide 
the  edge  is  straight. f 

175.  LEVELS  ON  ALIDADE.    The  bubble  should  be  in 
the  middle  when  the  table  is  level.     To  make  this  ad- 
justment, place  the  alidade  in  the  middle  of  the  table 
and  bring   the  bubble  to  the  center  by  means  of   the 
leveling  screws  of  the  table.     Draw  lines  along  the  edge 
of  the  rule  to  show  its  exact   position,  and  then  reverse 
it  180°.     If  the  bubble  remains  in  the  center,  it  is  in  ad- 
justment.    If  it  does  not,  correct  one  half  by  means  of 
the  leveling  screws  of  the  table,  and  the  other  half  by 
the  adjusting  screws  attached  to  the  level. 

It  is  next  to  impossible  to  make  this  adjustment  ac- 
curately if  the  table  is  not  a  perfect  plane.     Great  care 


*  For  general  remarks  upon  the  adjustments,  see  §  37. 

t  "  There  is  one  deviation  from  a  straight  line,  which,  by  a  very  rare  possi- 
bility, the  edge  of  the  ruler  might  assume,  and  yet  not  be  shown  by  the  above 
test.  It  is  when  a  part  is  convex,  and  a  part  similarly  situated  at  the  other 
end  concave,  in  exactly  the  same  degree  and  proportion.  In  this  case,  on 
reversal,  a  line  drawn  along  the  edge  of  the  rule  would  be  coincident  with  the 
other,  though  not  a  true  right  line."  To  determine  whether  this  defect  exists, 
move  the  alidade  endwise  and  draw  a  third  line. 


154  PLANE    TABLE.  [CHAP.  IX 

is  necessary  not  to  disturb  the  table  in  making  the  ad- 
justment. 

176.  BOARD,     i.  The  top    surface  should  be  a  plane. 
Test  it  in  all  directions  by  a  straight-edge  (§  174).     If 
it   is  not   perfectly  flat,  work   it  down   with   a   smooth 
plane,  a  scraper,  or  sand-paper. 

2.  The  face  of  the  table  should  be  perpendicular  to 
the  vertical  axis  of  the  instrument.  To  make  this  ad- 
justment, set  up  the  instrument,  place  the  alidade  on 
the  table,  and  bring  one  of  the  bubbles,  preferably  the 
one  on  the  telescope,  to  the  middle  ;  then  reverse  the 
table  on  its  axis.  If  the  bubble  has  not  moved,  the 
portion  of  the  table  covered  by  the  alidade  is  perpendic- 
ular to  the  vertical  axis  ;  if  it  has  moved,  correct  half 
the  error  by  inserting  washers  between  the  table  and 
the  arms  connecting  it  to  the  tripod  head. 

Turn  the  alidade  90°  on  the  face  of  the  board  and 
repeat  the  test. 

177.  TELESCOPE,     i.  The  line  of  sight  of  the  telescope 
should  be  perpendicular  to   the  horizontal   axis.     This 
adjustment    is   the   same   as    that    described   in   §   123 
(page  109). 

2.  The  horizontal    axis  of  the   telescope    should  be 
parallel  to  the  top  of  the  table.     This  is  the  same  ad- 
justment as  that  discussed  in  §  125  (page  in). 

3.  To  make  the  line  of  collimation  coincide  with  the 
fiducial  edge  of   the   alidade,  level  the  table,  set    two 
needles   in   its   face  in  range  with   some   object,  say  10 
feet    away;    then   place  the   fiducial   edge    against   the 
needles,  and  direct  the  telescope  toward  .the  object.     If 
the  cross  hairs  bisect  it,  the  adjustment  is  correct ;    but 
if  they   do   not,  it   can   be   corrected   by   means  of  the 
screws  attaching  the  standard  to  the  rule. 

The  line  of  sight  of  the  telescope  is  usually  in  the 
plane  of  the  fiducial  edge,  although  it  is  not  necessary 
that  it  be  either  in  the  plane  of  the  edge  or  parallel  to 


ART.  3]  USING    THE    PLANE    TABLE.  155 

it;  but  it  is  necessary  that  the  two  should  have  a  fixed 
horizontal  angle  with  each  other. 

178.  ZERO  OF  VERNIER.     The  vernier  of  the  vertical 
arc  should  read  zero  when  the  line  of  sight  is  horizontal. 
This  adjustment  is  the  same  as  that  of  §  127  (page  114). 
This  adjustment  is  important  when  elevations  are  to  be 
determined  by  vertical  angles. 

179.  LEVEL  ON  TELESCOPE.     The  bubble  should  be  in 
the  middle  when  the  line  of  sight  is  horizontal.     This  is 
the  same  adjustment  as  that  of  §  126  (page   112).     It 
is  important  only  when  elevations  are  to  be  determined 
by  using   the  telescope  as  a  level  and  measuring   the 
difference  of  heights  by  a  level  rod.     Since  the  level  on 
the  telescope  is  ordinarily  more  sensitive  than  the  plate 
levels,  difference  of  elevations  can  be  determined  more 
accurately  by  horizontal  lines  of  sight  than  by  vertical 
angles. 

ART.  3.     USING  THE  PLANE  TABLE. 

180.  For  many  kinds  of  work  the  plane  table  is  a  very 
useful  instrument.     Even  the  simplest  form  (§  172)  may 
be  used  to  advantage  in  obtaining  the  plat  and  area  of 
the  irregular  tracts  that  occur  in  ordinary  land  and  city 
surveying  ;  and  it   is  valuable  in  making  plats  of  parks, 
cemeteries,  mining  property,  etc.     It  has  the  great  ad- 
vantage of  dispensing  with  all  notes  and  records  of  the 
measurements  (since  they  are  platted  as  they  are  sur- 
veyed), and  thus  saves  time   and  lessens  the  possibility 
of  making  mistakes. 

181.  METHODS.     Points   may  be  located  with  respect 
to  each  other  by  any  of   four   methods,  viz.,  (i)  radia- 
tion, (2)  traversing,  frequently   called   progression,  (3) 
a  combination  of  the  first  and  second,  here  called  radio- 
progression,  and  (4)  intersection.     The   explanation  of 
these  methods  will  be  worded  for  the  determination  of 


PLANE    TABLE. 


[CHAP,  ix 


a  plat  of  a  field,  because  that  process  includes  all  the 
operations  involved  in  plane-table  work  with  one  excep- 
tion— viz.,  placing  the  table  in  position  at  an  undeter- 
mined point  by  observations  upon  three  known  points 
(see§  191). 

With  a  little  study  the  operator  can  discover  many 
modifications  and  combinations  of  these  methods.  The 
facility  with  which  plane-table  work  can  be  modified  to 
suit  circumstances  is  one  of  its  excellencies. 

182.  Radiation.  This  is  the  most  common  method  of 
using  the  plane  table,  particularly  in  topographical 
surveying.  It  is  the  simplest,  though  not  the  most 
accurate,  method  of  finding  the  plat  of  a  field. 

Let  A,  B)  C,  D,  E,  F,  Fig.  40,  represent  the  corners 


FIG. 


of  a  field  the  plat  of  which  is  to  be  found,  and  the  rect- 
angle near  its  center  the  plane  table.* 

Set  the  instrument  at  some  point  6>,  near  the  center  of 
the  field,  from  which  all  the  corners  are  visible.     Level 


*  The  size  of  the  table  in  this  and  the  following  figures  is  greatly  exagger- 
ated. 


OF  THK 

ART.  3]  USING    THE    PLANE  fcXHW^VERSHTY 


the  table,  and  stick  a  pin  or  needle  havrrrg^^eafing-wax 
head  in  th.e  prolongation  of  the  plumb-line.*  To  deter- 
mine whether  the  pin  is  set  exactly  right,  revolve  the 
board  and  notice  whether  the  pin  is  stationary.  Place 
the  alidade  against  the  pin,  direct  it  to  any  corner  of 
the  field,  as  A,  and  draw  a  line.  Measure  OA,  and  set 
off  this  distance,  to  any  convenient  scale,  from  the 
needle  along  the  line  just  drawn  to  a.  In  the  same  way 
plat  b,  c,  dt  etc.,  to  represent  the  corresponding  corners 
of  the  field.  Join  a  and  £,  b  and  c,  etc.,  and  a  complete 
plat  of  the  field  is  obtained. 

Notice  that  this  method  is  deficient  in  checks  upon 
the  accuracy  of  the  work.  Any  movement  of  the  table 
during  the  progress  of  the  work  may  be  detected  by 
sighting  upon  the  first  station. 

183.  The  position  of  trees,  houses,  etc.,  may  be  deter- 
mined in  the  same  way.     By  using  an  alidade  having 
a  telescope  with  stadia  hairs,  the  distance  to  the  object 
may  be  determined   from  the  stadia   reading  and   the 
elevation  from  the  vertical  circle  (see  Chapter  X.).    Thus 
the  map  may  be  drawn  in  the  field,  which  is  a  decided 
advantage  in  some  respects,  but  seriously  objectionable 
in  others. 

184.  Traversing.      This  method   is  frequently  called 
progression  /  but  as  it  is  strictly  analogous  to  traversing 
with  the  transit  (§  137),  the  term  traversing  seems  pref- 
erable.    Let  A  BCD  be  the  field  to  be  surveyed.     Select 
some  point,  a,  Fig.  41,  on  the  table  to  represent  the  first 
corner  of  the  field,  A.     Estimate  the  dimensions  of  the 
field  and    so  locate  a  that  the  plat  will  fall  upon  the 
board.     Draw  a  line  through  a  to  represent  AB\  then 
measure  AB,  and  lay  off  its  length  along  this  line  to  b. 


*  The  center  of  the  table  should  be  permanently  marked  by  setting  flush 
with  the  face  of  the  board  a  piece  of  brass,  say  one-fou'rth  inch  in  diameter, 
having  in  it  a  hole  just  large  enough  to  admit  a  common  pin. 


'58 


PLANE    TABLE. 


[CHAP,  ix 


Set  the  instrument  at  JSy  so  that  the  point  b  on  the 
board  shall  be  exactly  over  the  corresponding  point  on 
the  ground  and  the  line  ab  on  the  plat  shall  have  the 


FIG.  41. 


same  direction  as  AB  on  the  ground.  To  satisfy  this 
condition  with  any  considerable  degree  of  accuracy* 
requires  great  patience  and  care.  The  difficulty  is  that 
having  placed  the  point  on  the  board  over  the  point  on 
the  ground,  it  is  necessary  to  destroy  this  condition  in 
turning  the  board  to  make  the  direction  of  ab  coincide 
with  that  of  AB.\ 

Having  placed  b  over  the  corresponding  point  on  the 
ground  and  made  the  direction  of  ab  coincide  with  that 
of  AB,  place  the  alidade  against  the  needle  at  b  and 
sight  to  C.  Measure  the  distance  BC>  and  lay  it  off 


*  The  degree  of  accuracy  required  in  setting  the  point  on  the  board  over  the 
corresponding  point  the  on  ground  depends  upon  the  scale  of  the  map  and 
the  distance  to  the  object  sighted  at. 

t  A  plane  table  with  a  shifting  center  would  be  very  convenient  for  this  pur- 
pose ;  but  such  tables  are  not  made,  owing  to  mechanical  difficulties  in  their 
construction.  A  German  plane  table  is  provided  with  a  double  slide-rest 
motion  for  this  purpose. 


ART.  3]  USING    THE    PLANE    TABLE.  159 

from  b  to  c.  Move  the  instrument  to  C,  and  proceeds  as 
before.  At  the  last  station,  D,  determine  the  position 
of  the  first  station,  A,  as  though  it  were  not  already 
platted.  The  agreement  of  the  two  determinations  of 
a  is  a  check  upon  the  accuracy  of  the  work.  The  work 
may  be  checked  as  it  progresses,  by  seeing  whether  any 
line,  as  ca,  on  the  plat  agrees  with  the  corresponding 
line,  CA>  on  the  ground.  Tlxe^ahility-to  check  the  work 
at  every  step  is  a  valuable  feature  of  this  method. 

185.  Instead  of  trying  to  place  the  point  on  the  paper 
exactly  over  the  point  on  the  ground,  it  is  better  to  set 
the  table  level  and  approximately  over  the  point,  and 
then  sight  a  line  through  the  point  on  the  paper  to  a 
temporary  point  having  a  corresponding  position  with 
reference  to  the  point  to  be  determined.     For  example, 
if  in  setting  the  table  over  K  to   determine  the  direc- 
tion of  Z,  the  point  k  is  6  inches  perpendicularly  to  the 
right  from  the  line  KL,  set  near  L  a  point  6  inches  per- 
pendicularly  to  the   right    and  sight   at    the    point  so 
marked.     The  line :  drawn  on  the  paper  is  then  parallel 
to  the  line  on  the  ground.     The  true  distance  K  L  is  to 
be  measured  and  laid  off  on  the  paper. 

186.  This  method  is  especially  suited  to  the  survey 
of  a  road,  stream,  or  the  like.     Often  the  offsets  required 
may  be  sketched  in  by  the  eye  with  sufficient  accuracy. 

When  the  paper  is  filled,  put  on  a  new  sheet,  and  begin 
by  fixing  on  it  two  points  which  were  on  the  former 
sheet,  and  from  them  proceed  as  before.  The  sheets 
can  then  afterwards  be  united,  so  that  all  the  points 
shall  be  in  their  true  relative  positions. 

187.  Radio-progression.      This   method   combines  the 
simplicity  of  the  method  by  radiation  with   the  checks 
of  the  method  by  traversing  (progression).     The  chief 
advantage   is   the  convenience  with  which  the  table  is 
set  up   at  each   station,  since   the  center  of  the  table  is. 
always  set  over  the  station  occupied. 


i6o 


PLANE    TABLE. 


[CHAP,  ix 


Place  a  needle  in  the  center  of  the  board  and  set  the 
table  over  any  corner  of  the  field,  as  A,  Fig.  42.  Take 
a  sight  to  one  of  the  adjacent  corners,  as  E.  Next  sight 
upon  the  other  adjacent  corner,  as  B.  To  avoid  con- 
fusion, mark  the  lines  so  as  to  indicate  the  side  of  the 
field  to  which  they  correspond.  Move  the  table  to  the 
station  last  sighted  at,  set  it  up,  and  turn  it  until  the 
alidade,  when  placed  upon  the  last  line  drawn,  bears 
upon  the  station  previously  occupied.  Sight  at  the 
next  station  in  order  around  the  field,  and  proceed  in  a 
similar  manner  at  all  the  corners.  Fig.  42  represents 


FIG.  42. 

the  table  at  each  corner  in  succession,  commencing  at  A^ 
The  sides  of  the  field  and  the  corresponding  lines  on  the 
table  are  numbered  similarly  for  convenience  of  refer- 
ence. If  the  sighting  is  correctly  done,  the  fore-sight 
from  the  last  station  will  coincide  with  the  back-sight 
from  the  first  station.  The  work  may  be  checked  as  it 
progresses  by  sighting  at  some  other  than  the  two  adja- 
cent points  used  above,  ad,  Fig.  42,  is  such  a  check  line. 
The  lengths  of  the  several  sides  are  to  be  measured 
as  usual. 


ART.  3] 


USING    THE    PLANE    TABLE. 


161 


It  is  obvious  that  the  lines  radiating  from  the  center 
of  the  table  are  respectively  parallel  to  the  sides  of  the 
proposed  plat.  To  draw  the  plat,  it  is  only  necessary  to 
assume  an  initial  point  and  draw  lines  through  it  par- 
allel to  any  two  adjacent  sides  of  the  field,  and  then  lay 
off  on  these  lines  the  lengths  of  the  respective  sides, 
to  any  convenient  scale  (see  Fig.  43).  Two  other  cor- 


FlG.    43- 

ners  of  the  plat  are  thus  determined.  Through  each  of 
tnese  draw  a  line  parallel  to  the  next  side  of  the  field  ; 
and  do  likewise  for  all  the  remaining  corners. 

The  drawing  can  be  checked  as  it  progresses,  by  plat- 
ting the  check  lines  and  noticing  whether  they  pass 
through  the  corresponding  point  in  the  map.  The 
closing  of  the  plat  checks  the  accuracy  of  all  the  work. 

188.  The  principle  of  having  the  center  of  the  table 
always'over  the  point  occupied  may  be  applied  in  locat- 
ing the  position  of  trees,  corners  of  buildings,  points  on 
streams,  etc.,  by  drawing  through  the  point  on  the  table 
corresponding  to  the  point  over  which  the  instrument 
is  set  a  line  parallel  to  the  edge  of  the  alidade.  This 
requires  the  use  of  two  large  and  similar  triangles  in 


162  PLANE    TABLE.  [CHAP.  IX 

drawing  the  line  parallel  to  the  edge  of  the  alidade. 
The  point  occupied  by  the  instrument  being  located  on 
the  paper,  the  alidade  is  placed  against  the  needle  stand- 
ing in  the  center  of  the  board,  and  the  sight  turned 
upon  the  point  to  be  platted.  Then  without  disturbing 
the  alidade  place  one  triangle  against  the  edge  of  the 
alidade  and  the  second  triangle  against  the  first,  and 
slide  one  against  the  other  until  the  edge  of  the  second 
passes  through  the  point  on  the  paper  corresponding  to 
the  point  occupied.  Next  draw  a  line  along  the  edge 
of  the  triangle,  and  on  this  line  lay  off  the  distance  from 
the  instrument  to  the  point  sighted  at.  Proceed  in  like 
manner  for  each  point  to  be  located. 

Before  moving  the  table  to  the  next  station,  it  is  well 
to  check  the  position  of  the  table  by  re-determining  the 
direction  of  the  first  line.  For  greater  accuracy  and 
speed  in  making  this  test,  it  is  better  to  draw  a  line 
through  the  center  of  the  board  to  mark  the  first  position 
of  the  alidade.  Having  checked  the  position  of  the 
table,  sight  to  the  next  station  to  be  occupied,  and 
mark  the  position  of  the  alidade  by  a  line  through  the 
center  of  the  board.  This  line  must  also  be  transferred. 
Then  move  the  instrument  to  the  next  station,  set  the 
tripod  over  the  point,  level  the  board,  place  the  alidade 
on  the  last  line  drawn,  through  the  center  and  turn  the 
table  until  the  lineof  sight  covers  the  point  formerly 
occupied.  Observations  may  now  be  made  as  at  first. 

If  a  considerable  number  of  points  are  to  be  deter- 
mined at  each  setting  of  the  instrument,  it  is  easier  to 
spend  some  time  in  setting  the  point  on  the  paper  over 
the  corresponding  point  on  the  ground,  than  to  transfer 
all  the  lines;  but  if  there  are  only  a  few  points  to  be 
determined  at  each  setting,  it  is  easier  to  set  the  center 
of  the  board  over  the  point  and  transfer  the  lines 
One  advantage  of  keeping  the  center  of  the  board  over 
the  point  on  the  ground  is  that  the  alidade,  which  has 


ART.  3]  USING    THE    PLANE    TABLE.  163 

considerable  weight — particularly  one  carrying  a  tele- 
scope— always  stands  centrally  on  the  board  and  never 
near  one  edge.  Notice  that  the  two  triangles  take  the 
place  of  the  plumbing  bar.  As  ordinarily  made,  plane 
tables  have  no  means  of  hanging  a  plumb-line  from  the 
center  of  the  instrument;  but  the  deficiency  is  easily 
and  cheaply  supplied.  If  the  two  edges  of  the  alidade 
are  parallel,  the  triangle  may  be  placed  against  either 
edge,  as  is  most  convenient. 

189.  Intersection.  Measure  any  line,  preferably  cen- 
tral in  the  tract  to  be  surveyed,  on  the  ground,  as  M  JV, 
Fig.  44.  Select  some  point  on  the  paper  to  represent 


-4- 

*'» 


c/ 


FIG.  44. 


My  draw  a  line  through  it  to  represent  the  direction  of 
the  line  MN,  and  lay  off  the  distance  MN.  Then  set 
the  point  on  the  paper  representing  M  over  the  corre- 
sponding point  on  the  ground,  and  orient  the  table  so 
that  the  alidade  when  placed  upon  the  line  MN  will 
sight  to  N,  and  clamp  the  instrument.  Then  sight  to 
all  the  points  whose  location  is  desired,  as  A,  B,  C,  £>, 
etc.,  and  draw  lines  to  show  the  direction  of  these 
points.  Each  line  and  each  point  should  be  so  marked 
that  they  can  be  £ejj^hil^_identified.  Notice  that  if 


764  PLANE  TABLE.  [CHAP.  IX 

there  are  any  considerable  number  of  lines  it  is  not 
enough  simply  to  put  a  letter  or  figure  on  the  line,  as  in 
Fig.  44,  since  another  line  may  be  subsequently  drawn 
through  the  number.  Under  these  conditions  the 
designation  of  the  line  should  be  placed  in  a  small 
rectangle,  one  side  of  which  coincides  with  the  line  to 
be  identified.  If  necessary,  the  points  sighted  at  may 
be  marked  by  driving  a  numbered  stake  beside  them. 

Next,  set  the  instrument  over  N,  placing  the  point  on 
the  paper  over  the  corresponding  point  on  the  ground, 
and  orient  the  table  so  that  the  direction  of  the  line 
on  the  paper  corresponding  to  MN  coincides  with  that 
line  on  the  ground.  Then  sight  at  all  the  points  and 
draw  lines  to  correspond  with  their  new  directions. 
The  intersection  of  the  two  lines  of  sight  to  each  point 
will  determine  its  position. 

If  there  are  other  points  not  visible  from  M,  as  will 
be  the  case  in  a  line  survey,  sight  at  them  from  JV  and 
measure  a  new  base  line,  and  proceed  as  befcre.  Or 
the  end  of  the  new  base  line  may  be  determined,  as  any 
other  point,  by  tne  intersection  of  two  sight  lines  from 
the  ends  of  the  original  base  line.  Obviously,  measur- 
ing the  new  base  directly  is  the  more  accurate. 

190.  Before  the  introduction  of  the  stadia,  the 
method  of  intersection  was  the  most  usual  and  most 
rapid  method  of  using  the  plane  table.  But  this 
method  affords  no  check  upon  the  accuracy  of  the 
work,  is  often  deficient  in  precision  on  account  of 
oblique  intersections,  and  requires  so  many  lines  upon 
the  board  as  to  cause  confusion  and  error.  It  is  well 
adapted  to  the  mapping  of  harbors,  shore  lines,  and 
generally  to  the  plotting  of  inaccessible  points.  Of 
course,  in  this  as  in  all  triangulations,  weH^conditioned 
triangles  give  more  satisfactory  results;  or  in  other 
words,  avoid,  if  possible,  angles  less  than  30°  or  greater 
than  150°. 


ART.  3]  USING    THE    PLANE    TABLE.  165 

191.  THE  THREE-POINT   PROBLEM.     In  this  problem 

three  stations  A>  B,  C*  are  plotted,  as  a,  b,  c,  on  the 
table,  and  the  instrument  being  set  up  over  a  fourth 
point  Z>,  it  is  required  to  find  the  position  of  this  point 
on  the  map.  This  problem  is  indeterminate  when  the 
point  D  lies  in  the  circumference  of  a  circle  passing 
through  A,  B,  and  C,  in  which  case  the  two-point  prob- 
lem (§  194)  may  be  applied.  The  three-point  problem, 
which  also  occurs  in  the  use  of  the  sextant  in  locating 
soundings,  has  been  much  discussed,  and  many  solu- 
tions have  been  proposed. f  Only  two  will  be  given 
here. 

192.  Mechanical  Solution.     Fasten  a  she.et  of  tracing- 
paper  on  the  board,  and  fix  a  point  d  to  represent  the 
station  at  which  the  instrument  is  set.     With  the  ali- 
dade centring  on  </,  direct  the  telescope  successively  to 
A,  B,  and  C,  and  draw  lines  of  indefinite  length  along 
the  ruler's  edge  towards  these  stations.     Then,  if  the 
tracing-paper  be  shifted  until  the  three  lines  thus  drawn 
pass  through  the  points  «,  by  and  c,  the  point  d  will  indi- 
cate the  position  of  D.     The  position  of  this  point  may 
now  be   transferred   to  the  map  by  pricking   through. 
The    tracing-paper    is    then     removed,    and,  the    table 
oriented. 

193.  Graphical  Solution.  J     Let  a,  b,  and  <:,  Fig.  45,  be 
the  points  on  the  sheet  representing  the  signals  A,  B, 
and  C,  on  the  ground.     Set  the  table  at  the  point  to  be 
determined,  D,  and  level  it.     Set  the  alidade  upon  the 
line   ca,   and   by   revolving    the   table,   sight    upon    the 
signal  A,  and  clamp.     Then  with  the  alidade  centring 


*  The  capital  letters  refer  to  the  points  on  the  ground,  and  the  lower-case 
letters  to  the  corresponding  points  on  the  board. 

t  The  U.  S.  Coast  and  Geodetic  Survey  Report  for  1880,  pp.  180-84,  gives 
a  number  of  solutions  elaborately  illustrated. 

I  Known  as  Bessel's  method  by  inscribed  quadrilateral — see  U.  S.  Coast 
and  Geodetic  Survey  Report,  1880,  p.  181. 


j66 


PLANE  TABLE. 


[CHAP.  IX 


a 


on  <r,  sight  upon  the  middle  signal  B,  and  draw  the  line 
ce  along  the  edge  of  the  ruler.  Set  the  alidade  upon 
the  line  ac,  direct  the  telescope  to  the  signal  C  by 

revolving  the  table,  and 
clamp.  Then  with  the  ali- 
dade centring  on  «,  direct 
the  telescope  to  the  middle 
signal  B,  and  draw  the  line 
ae  along  the  edge  of  the 
ruler.  The  point  e  (the 
intersection  of  these  two 
lines)  will  be  in  the  line 
passing  through  the  mid- 
dle point  and  the  point 
sought.  Set  the  alidade 
upon  the  line  be,  direct  b 
to  the  signal  B  by  revolv- 
ing the  table,  and  the  table 
will  then  be  in  position. 
Clamp  it,  center  the  ali- 
dade upon  a,  direct  the  telescope  to  the  signal  .4,, and 
draw  along  the  ruler  the  line  ad.  This  will  intersect 
the  line  be  at  the  point  sought.  To  verify  its  position, 
center  the  alidade  on  f,  and  sight  to  C. 

The  opposite  angles  of  the  quadrilateral  adce  being 
supplementary,  the  angles  ace  and  ade  are  subtended  by 
the  same  chord  ae,  and  cae  and  cde  are  subtended  by 
the  same  chord  ce  ;  and,  consequently,  the  intersection 
of  ae  and  ce  at  e  must  fall  on  the  line  db.  Or,  the 
segments  of  two  intersecting  chords  in  a  circle  being 
reciprocally  proportional,  the  triangles  adf  and  cef  are 
similar,  as  also  the  triangles  cdf  and  aef\  and  there 
fore  </,/,  and  e  must  be  in  a  right  line  passing  through  £. 
194.  THE  TWO-POINT  PKOBLEM.  There  are  several 
solutions*  of  this  problem,  only  one  of  which  will  be 


FIG.  45. — THREE-POINT  PROBLEM 


*  U.  S.  Coast  Survey  Report,  1880,  pp.  184-85. 


ART.  3]  USING    THE    PLANE    TABLE.  167 

r\ 

given.  Two  points,  A  and  B,  not  conveniently  accessi- 
ble, being  given  by  their  projections  a  and  ^,  it  is  re- 
quired to  put  the  plane  table  in  position  at  a  third! 
point,  C.  Select  a  fourth  point,  D,  Fig.  46,  such  that 
the  intersections  of  lines  from  C  and  D  upon  A  and  B 
make  sufficiently  large  angles  for  good  determinations. 
Put  the  table  approximately  In  position  at  Z>,  by  esti- 
mation or  by  compass,  and  draw  the  lines  Aa  and  Bb, 

intersecting  in  d9 .     Through  d'     a\\^ [_ 

draw  a  line  directed   to  C,  and 

on  this  line  lay  off,  from  </',  the 

estimated     distance     CD,    and 

mark   the   point  thus  found  c'. 

Set  the  instrument  on  C  with  c' 

over  the  point,  and  orient  on  £> 

by   the    line    c'd'.      Draw    lines 

from  cr  to  A  and  to  B.     These 

will  intersect  the  lines  d' A  and 

d'B  at  points  a9  and  £',  which     FlG-  <6- -TWO-POINT  PROBLEM. 

form  with  c'  and  d'  a  quadrilateral  similar  to  the  true 

one,  but  erroneous  in  size   (since  the  distance  c'd9  was 

assumed),    and    in    position    (since    the    table   was   not 

properly  oriented  at  either  station). 

The  angles  which  the  lines  ab  and  afb'  make  with 
each  other  is  the  error  in  position.  By  constructing 
now  through  c'  a  line  ?a*t  making  the  same  angle  with 
cfdf  as  that  which  ab  makes  with  a'b'  and  directing  the 
line  c'd'  to  D,  the  table  will  be  brought  into  position, 
and  the  true  point  c  can  be  found  by  the  intersection  of 
a  A  and  bB. 

Instead  of  transferring  the  angle  of  error  by  construc- 
tion, we  may  conveniently  proceed  as  follows,  observ- 
ing that  the  angle  which  the  line  a'b'  makes  with  ab  is 
the  error  in  the  position  of  the  table.  As  the  table  now 
stands,  a'b9  is  parallel  with  AB ;  but  we  want  to  turn  it 


l68  PLANE    TABLE.  [CHAP.  IX 

so  that  ab  shall  be  parallel  to  that  line.  Therefore  if 
we  place  the  alidade  on  a'b'  and  set  up  a  mark  in  that 
direction,  and  then  place  the  alidade  on  ab  and  turn 
the  table  until  it  again  points  to  the  mark,  ab  will  be 
parallel  to  AJ3,  and  the  table  is  in  position. 

195.  SOURCES  OF  ERROR.*  The  sources  of  error  in 
plane  table  work,  in  addition  to  those  of  chaining 
(§  19)  or  stadia  measurements  (§§229-32)  are  as  follows: 
(i)  error  of  position  of  instrument;  (2)  error  of  sighting; 
(3)  movement  of  the  board  between  sights;  (4)  errors 
of  adjustment  of  the  instrument;  (5)  the  inclination  of 
the  board;  (6)  error  in  marking  the  line  upon  the 
paper;  (7)  error  in  scaling  off;  and  (8)  the  hygrometric 
effect  of  the  atmosphere  on  the  paper.  The  effect  of 
any  of  these  errors  will  depend  upon  the  scale  of  the  map. 
The  first  four  are  essentially  the  same  as  the  corre- 
sponding errors  in  transit  work,  for  discussion  of  which 
see  §  140.  If  the  board  is  not  level,  an  error  is  pro- 
duced in  determining  horizontal  angles  between  points 
not  in  the  same  horizontal  plane,  and  also  in  determin- 
ing vertical  angles  between  points  not  in  the  same  ver- 
tical plane.  The  sixth  and  seventh  depend  upon  the 
skill  and  care  of  the  draughtsman.  The  changes  in  the 
dimensions  of  the  drawing,  due  to  the  hygrometric 
state  of  the  atmosphere,  will  vary  with  the  weather  and 
the  kind  of  paper.  The  U.  S.  Coast  Survey,f  to  de- 
termine this  variation,  cut  several  strips  two  meters 
long  from  three  samples  of  drawing  paper,  and  ob- 
served the  variations  daily  for  six  months.  For  strips 
cut  longitudinally  the  average  variation  was  9.0  milli- 
meters, 13.0  millimeters,  and  15.7  millimeters,  and  the 
maximum  was  n.i,  15.5,  and  20.9,  respectively;  and  for 


*  For  a  discussion  of  Compensating  vs.  Cumulative  Errors,  see  §  18. 
t  U.  S.  Coast  Survey  Report  for  1862,  p.  255. 


ART.  3]  USING    THE    PLANE    TABLE.  169 

strips  cut  transversely  the  average  variation  was  8.4,  8.1^ 
and  12. i  millimeters,  and  the  maximum  was  10.3,  9.6,  and 
15.0,  respectively. 

196.  LIMITS   OF    PRECISION.     Since   there  is   so  much 
variety  in  the  kind  of  work,  in  the  method  of  doing  it, 
and    in    the    conditions  under  which    it   is   done,   it    is 
scarcely  possible  to  state  in  general  the  degree  of  pre- 
cision   to  be  obtained  with  the  plane  table.     Further- 
more the  plane  table  is  designed  more  for  rapidity  than 
accuracy,   although   the  best   modern   instruments  are 
capable  of  a  considerable  degree  of  precision. 

197.  With  Plain  Alidade.     In  ordinary  class  work  of 
the  first  term  of  surveying  the  author's  students  meas- 
ured three  angles  of  a  triangle  and  four  angles  around 
a  point,  using  a  wooden  alidade  with  slit  and  string  for 
sights,  with   an   average  error,  for  the  best  thirty-two 
out  of  thirty-six  results,  of  i  minute  and  44  seconds  per 
angle   (a   probable  error    of    88    seconds).     Under    fair 
conditions   the   maximum   error   per  angle   should    not 
exceed  3  minutes. 

Under  the  above  conditions  the  error  of  finding 
areas*  was  as  follows:  for  radiation  (§  182),  an  average 
error  of  i  in  586  and  a  maximum  error  of  i  in  274;  for 
traversing  (§  184),  an  average  error  of  i  in  826  and  a 
maximum  error  of  i  in  450;  and  for  radio-progression 
(§  187),  an  average  of  i  in  1,111  and  a  maximum  error 
of  i  in  640.  For  the  sake  of  comparison  it  may  be 
interesting  to  know  that  for  the  same  students  under 
the  same  conditions  the  average  error  with  a  magnetic 
compass  (§  51)  was  i  in  1,220  and  the  maximum  i  in 
580;  and  with  a  chain  alone,  the  average  error  was  i  in 
1,520  and  the  maximum  i  in  500. 

198.  With  Telescope  Alidade.     Rarely,  if  ever,  would 
the  most  elaborate  form  of  plane  table  be  employed  in 

*  See  second  foot-note,  p.  28. 


170  PLANE    TABLE.  [CHAP.  IX 

finding  areas;  and,  for  reasons  before  stated,  it  is  im- 
possible to  give  a  summary  of  the  precision  attainable 
with  this  form  of  plane  table  in  topographical  survey- 
ing. Therefore  this  subject  will  be  concluded  with  a 
few  references  to  descriptions  of  surveys  made  with  the 
plane  table,  which  contain  data  on  precision,  cost,  and 
speed. 

1.  Professional   Papers  of  the  Engineer  Department, 
U.  S.  A.,  No.   18,— Report  of  the  Geological   Explora- 
tion of  the  Fortieth  Parallel,  by  Clarence  King, — Wash- 
ington, D.  C.,  1878,  Vol.  I,  p.  762. 

2.  U.  S.  Geographical  Surveys  West  of  icoth  Meridian, 
Wheeler,  1883,  p.  47. 

3.  Report  on   the  Third  International  Geographical 
Congress   and    Exhibition  at  Venice,   Italy,   1881,  Geo. 
M.  Wheeler,  Washington,  1885,  pp.  79-81. 

4.  Science  (New  York  City),  July  29,  1887,  p.  49. 

5.  Plane  Table  Methods  used  by  the  U.  S.  Geological 
Survey  in  Western  Massachusetts,  Louis  F.   Cutter,  in 
Journal    of    the    Association    of    Engineering    Societies 
(Chicago),  Vol.  X,  pp.  356-69. 

199.  PRACTICAL  HINTS.  The  board  should  be  placed 
so  low  as  to  be  readily  reached,  even  at  the  most  remote 
corners,  and  yet  high  enough  to  enable  the  observer 
to  sight  with  comfort.  This  will  bring  it  a  little  below 
the  elbow.  All  beginners  are  apt  to  set  the  table 
too  high.  Care  must  be  taken  that  no  part  of  the  body 
touch  or  rest  against  the  edge  of  the  board.  In  using 
the  alidade,  steady  the  standard  with  the  left  hand, 
while  the  right  swings  the  rear  end  of  the  ruler  in  the 
proper  direction. 

Manilla  paper  is  easier  on  the  eyes  in  the  sunshine 
than  white,  and  also  shows  dirt  less.  Thumb-tacks 
and  rollers  for  holding  down  the  sheet  are  both  ob- 
jectionable, especially  in  high  winds.  The  edges  may 
be  pasted  underneath,  or  spring  clamps  may  be  used 


ART.  3]  USING    THE    PLANE    TABLE.  17 1 

to  advantage.  The  sides  of  the  sheet  where  they  are 
turned  under  the  table  and  come  more  or  less  in  contact 
with  the  coat  of  the  observer,  should  be  protected  by 
strips  of  paper  about  4  inches  wide,  and  6  inches 
longer  than  the  sides  of  the  table,  so  as  to  fold  under  it 
and  clamp  on  with  the  sheet  itself.  Tracing  vellum  is 
good  for  this  purpose,  as  the  points  near  the  edge  of  the 
sheet  can  be  seen  through  it. 

A  short  linen  coat  with  large  pockets,  in  the  style  of 
a  hunting-coat,  probably  affords  the  best  means  for 
carrying  the  requisite  accessories  for  plane-table  work; 
viz.,  scale,  triangles,  pencils,  rubber,  note-book,  etc.  By 
this  means  the  weight  of  these  articles  is  distributed  in 
separate  pockets,  and  they  are  always  at  hand  when 
needed. 

On  beginning  the  work,  set  off  at  some  point  near 
the  middle  of  the  sheet,  the  magnetic  meridian  for  the 
purpose  of  putting  the  table  in  approximate  position  at 
any  subsequent  station  with  the  declinator. 

Use  as  hard  a  pencil,  and  make  as  few  lines,  as  possi- 
ble. In  locating  points  on  contours,  plot  the  distance 
at  once  along  the  edge  of  ruler  by  a  detached  scale, 
making  only  a  dot  at  the  point  which  should  receive 
the  number  of  the  contour.  Objects  on  a  straight  line 
may  be  quickly  located  by  plotting  the  ends  of  the  line 
and  determining  the  intermediate  points  by  the  inter- 
sections of  this  line  and  lines  of  sight  to  the  several 
points. 

Always  before  leaving  a  station,  and  also  at  intervals 
when  not  otherwise  employed,  take  sights  to  determine 
whether  the  board  has  been  displaced. 

It  is  well  to  have  ready  a  light  india-rubber  cloth 
cover  to  slip  over  the  board  in  case  of  a  sudden  shower, 
as  well  as  to  protect  the  paper  from  dust  on  the  roads, 
mud  in  swampy  ground,  etc.  A  metal  chart  case 
should  always  accompany  the  table  to  protect  the 


172  PLANE    TABLE.  [CHAP.  IX 

sheet  from  sudden  rain  and  other  injury  liable  to  occur 
in  the  transportation  of  the  sheet  to  and  from  the  field, 
and  for  its  safe  keeping  when  not  in  use.  Its  diameter 
should  be  not  less  than  3  inches,  for  no  sheet  can  be 
rolled  to  a  less  diameter  without  serious  rupture  of  the 
fiber  of  the  paper. 


CHAPTER   X. 
TELEMETERS. 

200.  TELEMETER    is   a    term   variously    employed    to 
designate  some  form  of  an  instrument  for  determining 
distances  by  means  of  the  visual  angle  subtended  by  a 
short  base.     There  are  a  great  number  of  instruments 
of  this  class,  but  the  only  ones  of  any  considerable  value 
in  engineering  practice  are  the   stadia  and  the  gradi- 
enter.     These  instruments  will  be  discussed  at  consid- 
erable length  in  Arts,  i  and  2,  respectively,  and  in  Art.  3 
various  forms  of  telemeters  will  be  briefly  described. 

ART.  1.     THE  STADIA. 

201.  The  stadia  is  an  instrument  for  determining  the 
distance   of   a  point  from  the    observer  by  the   visual 
angle  subtended  -by  an  object  of  known  size  placed  at 
the  point.     Ordinarily   not  only  the  distance  but  also 
the  horizontal  and  vertical  angles  are  observed,  these 
three  being  sufficient  to  determine  the  direction,  dis- 
tance, and  elevation  of  the  point  upon  which  the  rod  is 
placed.     In  1820  Parro,  an  Italian  engineer,  first  sug- 
gested the  determination  of  distances  in  surveying  by  a 
visual  angle  and  a  rod.*     He  used  the  word  stadia  to 
designate  the  rod,  but  the  term  is  now  generally  ap- 
plied to  the  instrument  as  a  whole.     On  the  U.  S.  Coast 
and    Geodetic    Survey    the    term    telemeter   is    used    in- 
stead of   stadia.     In   Great  Britain   the   instrument  by 
which  the  observation  is  made  is  called  a  tacheometer, 
and  the  rod  is  called  a  stadia. 

*  William  Green,  a  London  optician,  in  1778  published  a  pamphlet  giving 
a  very  full  account  of  the  stadia,  but  the  credit  of  its  practical  introduction 
seems  to  be  due  to  Parro. 

173 


TELEMETERS.  [CHAP,  x 


The  principles  involved  have  long  been  well  known, 
and  have  been  applied  in  gunnery  and  military  recon- 
noissance  ;  but  it  is  only  lately  that  they  have  been 
used  in  engineering.  The  first  notable  use  of  the  stadia 
was  in  1836.,  in  making  a  topographical  survey  of 
Switzerland.  It  seems  not  to  have  been  introduced 
into  America  until  nearly  thirty  years  afterward,  and 
has  not  been  employed  here  to  any  considerable  extent 
except  in  topographical  surveys  carried  on  by  the  U.  S. 
Government.  It  has  not  come  into  as  general  use  as 
its  merits  warrant,  although  it  is  more  generally  used 
in  Europe  than  in  the  United  States.  The  stadia  is 
peculiarly  adapted  to  topographical  surveying,  for  it 
possesses  the  double  advantage  of  giving  both  the  hori- 
zontal and  vertical  co-ordinates,  and  this,  too,  by  the 
most  rapid  method. 

202.  PEINCIPIES.  In  all  its  forms  the  stadia  is  an  ap- 
plication of  the  principle  of  similarity  of  triangles.  The 
simplest  form  of  the  instrument  is  used  in  the  familiar 
method  of  determining  the  distance  of  a  man  from  an 
observer,  by  measuring  on  a  rule  held  at  arm's  length 
the  space  covered  by  his  height.  There  are  several 
forms  of  this  simple  device,  but  none  of  them  are  of 
any  practical  value  in  surveying,  owing  to  the  impossi- 
bility of  focusing  the  eye  for  two  distances  at  the  same 
time,  and  to  the  indistinctness  of  the  farther  object. 

To  employ  the  principle  of  the  stadia  with  a  tele- 
scope, it  is  necessary  to  introduce  two  parallel  cross 
hairs  and  observe  the  amount  of  the  rod  intercepted 
between  them.  The  hairs  may  be  horizontal  and  the 
rod  vertical,  or  vice  versa;  but  the  former  is  usually  pre- 
ferred, since  the  rod  is  then  more  steady,  and  there  is 
less  liability  of  its  being  obscured  by  brush,  etc.  The 
hairs  may  be  fixed,  the  intercept  on  the  rod  being 
variable  ;  or  the  hairs  may  be  movable  and  be  set  to 
cover  always  the  same  two  points  on  the  rod,  the  vari- 


ART.    l]  THE    STADIA.  175 

able  distance  between  the  hairs  being  measured  by  a 
graduated  screw.  The  fixed  hairs  are  cheaper,  more 
simple,  more  accurate,  and  in  every  way  better  than 
the  movable  ones  ;  and  are  generally  used. 

In  the  stadia  with  a  telescope  the  two  similar  trian- 
gles have  a  common  apex  at  the  optical  centre  (see  first 
foot-note,  page  106)  of  the  objective,  the  base  of  one 
being  the  distance  between  the  cross  hairs,  and  that  of 
the  other  the  intercept  on  the  rod.  Since  in  focusing 
the  telescope,  the  distance  from  the  cross  hairs  to  the 
objective  varies  with  the  distance  of  the  object  sighted 
at,  the  relation  between  the  distance  to  the  rod  and  the 
intercept  is  not  as  easily  found  as  with  the  simple  de- 
vice previously  mentioned.  The  formula  for  the  stadia 
with  telescope  will  be  deduced  presently. 

203.  PLACING  THE  HAIRS.  In  placing  the  stadia  hairs 
in  the  telescope  three  conditions  must  receive  attention  : 
they  should  be  parallel,  equally  distant  from  the  cen- 
tral one,  and  at  a  suitable  distance  from  each  other. 

These  conditions  are  the  rigorous  ones,  but  farther  on 
it  will  be  shown  that  no  appreciable  error  will  be  pro- 
duced if  they  are  only  approximately  satisfied.  It  is 
necessary  that  the  hairs  should  be  parallel  only  in  case 
the  observation  is  not  made  exactly  on  the  vertical  hair. 
If  the  stadia  hairs  are  not  equally  distant  from  the 
central  one,  it  produces  only  a  small  error  in  the  vertical 
angle.  Finally,  the  distance  between  the  hairs  is  im- 
material, provided  the  rod  is  graduated  to  correspond. 
A  formula  will  be  deduced  to  meet  the  case  in  which  the 
rod  is  already  graduated  and  does  not  agree  with  the 
distance  between  the  hairs  (§  215). 

Any  measuring  telescope  can  be  used  as  a  stadia  by 
adding  a  second  horizontal  hair  ;  but  it  is  much  better 
to  add  two  extra  ones,  one  on  each  side  of  the  ordinary 
one.  With  the  stadia  hairs  thus  placed,  the  field  of 
view  is  symmetrical  about  the  center,  and  the  added 


i76 


TELEMETERS. 


[CHAP,  x 


hairs  interfere  less  with  the  ordinary  uses  of  the  tele- 
scope. 

204.  Instrument  makers  not  infrequently  fasten  three 
horizontal  cross  hairs  on  the  reticule  (§  80),  the  two 
outer  ones  being  at  such  a  distance  apart  that  each  foot 
of  the  intercept  on  the  rod  corresponds,  at  least  approx- 
imately, to  a  hundred  feet  of  the  distance  from  the 
instrument  to  the  rod.  The  space  between  these  hairs 
is  computed  and  fine  grooves  are  engraved  at  the  proper 
distance,  into  which  the  hairs  are  laid  and  fastened.  If 
these  hairs  get  broken  the  engineer  can  replace  them 
(§  81)  with  a  little  care. 

Fig.  47  shows  a  common  method  of  making  the  dis- 
tances between  the  stadia  hairs  adjustable.  The  upper 
portion  of  Fig.  47  shows  the  reticule 
in  place  in  the  telescope  tube,  and 
the  other  shows  details.  The  screws 
d  and  f  can  be  operated  from  the 
outside  of  the  telescope  tube,  and 
move  (with  reference  to  the  ring  a) 
the  slides  b  and  c,  which  carry  the 
stadia  hairs.  The  screws  d  and  / 
are  shown  in  Fig.  22,  page  92,  im- 
mediately in  front  of  the  ordinary 
cross-hair  screws,  e  is  a  bent  spring 
to  take  up  lost  motion  in  the  screws 
d  and  /.  The  ring  a  and  all  parts 
connected  therewith  are  adjusted  in 
the  telescope  tube  by  the  ordinary 
cross-hair  screws,  which  are  not 
shown.  By  means  of  the  screws  d 
and/,  the  distance  of  the  stadia  wires 
from  the  central  wire,  and  from  each 
other,  can  be  varied  at  will.  Lines 
are  made  upon  the  slides  b  and  c  to  assist  in  placing  the 
hairs  parallel  to  each  other.  The  following  objections 


ART.    l] 


THE    STADIA. 


77 


are  sometimes  offered  to  this  construction  :  i.  The  pro- 
jecting heads  of  the  screws  d  and  /  are  liable  to  be 
struck  and  turned  in  handling  the  instrument  or  in 
carrying  it  through  brush,  and  so  produce  a  serious 
error  with  no  adequate  means  of  detecting  it.  2.  Since 
a  spring  must  be  inserted  to  take  up  lost  motion,  the 
screws  have  a  tendency  to  work  loose.  3.  It  is  expensive. 
205.  Fig.  48  shows  an  inexpensive  method  of  in- 
serting adjustable  .stadia  hairs,  in  an  instrument 
provided  with  them.  The  diagram 
represents  the  fro.nt  view  of  the  ordi- 
nary reticule,  a,  b,  c,  and  d  are  small 
wire  plugs  which  are  free  to  turn,  be- 
ing held  only  by  friction.  The  dark 
portion  is  in  the  plane  of  the- face  of 
the  ring,  and  the  light  portion  projects, 
say  an  eighth  of  an  inch,  above.  The 


not 


FIG.    48. 

hairs  are  to  be  stretched  in  the  line  of  the  centers  of  a 
and  c,  and  d  and  ^,  and  fastened  at  the  outer  edge  of  the 
ring.  Then  by  turning  the  plugs  the  hairs  will  be  moved 
toward  or  from  the  central  hair,  according  to  which 
side  of  the  plug  is  toward  it.  The  wire  plugs  may  easily 
be  made  to  fit  so  tight  as  not  to  work  loose  and  still 
turn  freely  enough  for  the  above  adjustment.  With 
telescopes  having  inverting  eye-pieces,  which  are  much 
the  best  for  stadia  work,  the  wires  can  be  turned,  after 
removing  the  eye-piece,  without  taking  the  reticule  out 
of  the  telescope  tube.  With  an  erecting  eye-piece,  the 
hairs  can  be  adjusted  without  removing  the  ring  from 
the  telescope  tube,  by  the  use  of  a  little  wrench  (made 
especially  for  this  purpose  by  cutting  a  kerf  with  a  hack- 
saw in  the  end  of  a  small  strip  of  brass),  and  operating 
it  through  a  hole  in  the  telescope  tube  (also  made  for 
the  purpose),  which  can  be  closed  by  a  metal  or  rubber 
band  ;  or  the  ring  may  be  removed  from  the  tube  to 
adjust  the  hairs.  The  telescope  does  not  need  to  be 


TELEMETERS.  [CHAP.   X 


collimated   to   test   the  relative   position    of  the   stadia 
hairs. 

This  method  provides  a  way  of  making  the  hairs 
parallel  to  each  other,  and  allows  a  variation  in  the  dis- 
tance between  the  central  and  each  outside  hair  equal 
to  the  diameter  of  the  plug,  which  may  be  made  of  any 
size  to  suit  the  circumstances. 

206.  The    maximum    distance    between    the   hairs  is 
limited  by  the  size  of  the  field  of  view  (§  87).     The  field 
of  view  of  the  telescopes  on  ordinary  engineering  instru- 
ments differs  with  different  makers,  but  is  about  as  fol- 
lows :  for  a  magnification  of  twenty,  i°  30';  of   twenty- 
five,  i°  15';  of  thirty,  i°;  and  of  thirty-five,  50'.     With 
the   smallest  power   mentioned   above  the  distance  be- 
tween  the    stadia  hairs  can  not    be    greater    than  one 
fortieth  of  the  focal  length  of  the  objective,  and  for  the 
largest  power  one  seventieth.     Since   it  is  not  possible 
to  have  the  hairs  at  the  extreme  edge  of  the  field  of  view, 
and  since  the  outer  portion  of  the  field  is  not  as  good 
optically  as   the  central  portion,  it   is   safe  to   say  that 
with  the  higher  powers  the  distance  between  the  stadia 
hairs  should  not  be  more  than  a  hundredth  of  the  focal 
length   of  the  objective.     With   the  lower   powers   the 
distance  between   the  hairs   might   be  greater,  but   for 
long  sights  the  rod  would  be  inconveniently  long. 

On  the  other  hand,  if  the  hairs  are  very  close  together 
the  intercept  on  the  rod  will  be  too  small  to  be  deter- 
mined accurately.  All  things  considered,  it  is  probably 
best  to  make  the  distance  between  the  hairs  a  hundredth 
of  the  focal  length  of  the  objective.  A  method  of  accu- 
rately making  this  adjustment  will  be  described  pres- 
•ently  (§§  214-15). 

207.  THE  ROD.      In  stadia  work  there  are  two  kinds  of 
rods  —  self-reading  and  target.     A  self-reading  rod  is  one 
having  a  graduation  such  that  the  value  of  the  intercept 
can  be  read  through  the  telescope.     A  target  rod  is  one 


ART.   l]  THE    STADIA.  179 

having  a  sliding  target  moved  by  the  rod-man,  in  re- 
sponse to  signals  from  the  instrument-man,  until  it  is 
in  the  plane  of  sight,  when  its  position  is  read  by  the 
rod-man.  The  latter  rod  requires  two  targets,  one  of 
which  is  sometimes  permanently  fixed  to  the  rod  to  save 
the  trouble  of  setting  two  targets  ;  but  when  the  vertical 
co-ordinate  is  desired,  this  adds  more  complication  than 
it  saves.  Target  rods  may  be  a  little  more  accurate,  but 
they  are  certainly  very  much  less  convenient.  Self- 
reading  stadia  rods  are  generally  preferred.* 

Figs.  49-52  (page  180)  show  a  few  of  the  many  gradua- 
tions proposed  for  self-reading  stadia  rods.  Fig.  49  is 
a  graduation  formerly  much  used  on  the  U.  S.  Coast 
and  Geodetic  Survey,  and  is  on  the  whole  the  best  of 
the  four  shown.  The  distance  from  a  to  b  corresponds 
to  10  feet  on  the  ground.  By  dividing  the  oblique  side 
of  the  triangle  into  fifths  by  estimation,  the  rod  may  be 
read  to  single  feet.  Notice  that  the  division  of  the 
oblique  side  of  the  triangle  is  the  principle  of  the  well- 
known  diagonal  scale.  Fig.  50  was  used  on  the  late 
U.  S.  Lake  Survey,  and  is  the  standard  on  the  surveys 
conducted  by  the  Mississippi  River  Commission.  It  is 
better  for  short  distances  than  Fig.  49,  but  not  so  good 
for  long  ones.  The  other  figures  are  added  to  show 
the  facility  with  which  designs  may  be  made.  Fig.  52  is 
very  good  for  short  distances,  and  very  poor  for  long 
ones.  As  a  rule  the  graduation  on  self-reading  stadia 
rods  is  not  numbered,  since  it  is  generally  considered 
easier  to  count  the  divisions  included  in  the  visual  angle 
than  to  read  both  hairs  and  subtract. 

Any  self-reading  leveling  rod  (§  270)  may  be  used  as 
a  stadia  rod  ;  but  as  a  rule  self-reading  leveling  rods 
have  a  great  number  of  very  small  divisions,  which  be- 
come indistinct  or  wholly  invisible  at  long  distances  ; 

*  For  a  discussion  of  the  relative  merits  of  self-reading  and  target  leveling 
rods,  see  §  270. 


8o 


TELEMETERS.  [CHAP.  X 


FIG,  49.  FIG.  50.  FIG.  51 


ART.    l]  THE    STADIA.  l8l 

and  hence  most  self-reading  leveling  rods  are  more  suit- 
able for  short  than  for  long  sights.  Leveling  rods  are 
generally  read  only  at  short  distances,  while  stadia  rods 
are  frequently  read  at  long  distances. 

208.  Apparently  stadia  rods  are  not  sold  by  instru- 
ment makers  ;  but  the  engineer  can  easily  make  his 
own. 

The  rod  must  be  light  and  convenient  for  transporta- 
tion. Usually  it  is  a  board  about  i  inch  thick,  4  or  5  inches 
wide,  and  from  10  to  14  feet  long.  It  may  be  stiffened 
by  screwing  a  strip  edgewise  on  the  back.  It  may  be 
hinged  in  the  middle,  and  folded  for  ease  of  transporta- 
tion and  for  the  protection  of  the  graduation  ;  and  when 
in  use,  it  may  be  kept  upright  by  a  button  or  a  bolt  on 
the  back.  The  rod  should  be  provided  with  a  disk 
level,  or  a  short  plumb,  for  keeping  it  vertical,  and  a 
handle  by  which  to  hold  it.  The  graduation  will  keep 
cleaner  and  last  much  longer  if  the  face  of  the  board 
is  recessed  slightly  to  receive  it.  The  face  should  be 
made  perfectly  white  by  a  number  of  thin  coats  of  paint, 
each  being  thoroughly  dry  before  the  succeeding  one 
is  applied. 

In  graduating  a  stadia  rod,  visibility  is  of  the  first 
importance.  If  the  graduation  consists  of  a  number  of 
small  divisions,  they  will  become  invisible  at  long  dis- 
tances, and  even  at  short  distances  will  be  very  confus- 
ing. The  graduation  marks  should  therefore  be  made 
large  and  the  smaller  divisions  be  obtained  by  sub- 
dividing the  larger  ones  by  estimation.  This  method 
by  estimation  is  almost,  or  quite,  as  accurate  as  when 
the  smaller  divisions  are  marked.  It  is  often  claimed 
that  very  small  subdivisions  can  be  read  more  exactly 
by  estimation  than  by  a  direct  graduation,  owing  to  the 
liability  of  error  in  counting  the  graduation  marks. 

The  pattern  may  be  painted  or  stenciled  directly  upon 
the  wood,  or  it  may  first  be  drawn  or  painted  upon 


l82  TELEMETERS.  [CHAP.  X 

paper  and  then  fastened  on  the  rod  with  varnish  or  any 
glue  not  soluble  in  water.  Stencils  may  be  made  of 
writing  paper  which  has  been  varnished  or  oiled.  In 
either  case  the  pattern  should  have  a  sharp  outline,  and 
should  be  marked  with  jet-black  paint  that  dries  with  a 
dead,  and  not  a  shiny,  surface. 

209.  To  determine  the  vertical  co-ordinates  of  the 
point  on  which  the  rod  is  set,  it  is  necessary  that  the 
line  of  sight  should  be  directed  to  a  point  at  a  known 
distance  from  the  foot  of  the  rod.  The  point  at  which 
the  central  visual  ray  is  directed  is  most  easily  indi- 
cated by  a  target,  which  may  be  either  fixed  or  movable, 
although  the  movable  target  is  much  the  better.  It 
may  be  made  of  a  piece  of  rolled  brass  2\  to  4  inches 
wide  and  about  an  eighth  of  an  inch  thick,  bent  as 
shown  in  Fig.  53.  The  pieces  a  and  b  are  made  concave 


F 


FIG.  53. 

on  their  inner  faces  to  fit  the  edges  of  the  rod.  a  is 
soldered  or  riveted  to  the  body  of  the  target,  and  b  is 
held  in  position  by  two  plates  (not  shown  in  Fig.  53) 
attached  to  the  top  and  bottom  of  it,  which  extend 
over  the  edges  of  the  body  of  the  target.  The  target  is 
clamped  at  any  point  on  the  rod  by  the  milled-head 
screw.  A  diamond  is  painted,  preferably  in  red,  on  the 
face  of  the  target  (see  §  268).  The  back  is  left  open  so 
the  target  may  be  moved  up  and  down  past  the  handle 
and  plummet  on  the  back  of  the  rod. 
210.  FORMULA  FOE  HORIZONTAL  LINE  OF  SIGHT  AND 

VERTICAL  ROD.  In  Fig.  54  let  a  and  b  represent  the 
stadia  hairs  ;  /  the  distance  between  them  ;  s  the  dis- 
tance,/^, on  the  rod  intercepted  between  the  hairs  ;  / 


ART.  i]  THE  STADIA*  183 

the  principal  focal  distance  of  the  objective  ;  e  a  point 
at  a  distance /in  front  of  the  optical  center  of  the  ob- 
jective, that  is,  e  is  the  principal  focus  of  the  objective  ; 
c  the  distance  from  the  plumb-line  of  the  instrument  to 
the  optical  center  of  the  objective  ;  y  the  distance  from 
the  outer  focus,  ^,  to  the  rod  ;  and  D  the  distance  from 
the  instrument  to  the  rod.  For  convenience  in  print- 
ing represent— .  by  k. 


FIG.  54. 

From  the  principles  of  optics,  we  know  that  all  rays 
of  light  which  pass  through  e  are  parallel  to  each  other 
after  emerging  from  the  objective.  Therefore  there  is 
some  point  ^,  which  will  emit  a  single  ray  of  light  that 
will  pass  through  e,  and,  after  traversing  the  objective 
will  strike  the  cross  hair  a.  If  the  telescope  is  focused 
for  the  point  ^,  the  objective  will  bring  all  rays  emitted 
by  q  to  a  focus  at  a  ;  and  hence  it  is  immaterial  whether 
we  consider  the  real  course  of  the  rays,  or  assume  that 
all  the  light  from  q  passes  along  the  line  qea. 

Similarly  we  may  assume  that  all  the  rays  from/  pass 
along  the  line  peb . 

From  Fig.  54  we  easily  get  s  \ y  ::/:/,  from  which 


(0 

rto 
each  instrument,  and  also  that  the  intercept  s  on  the 


Notice  that  k  =  4,  is  a  constant  coefficient  peculiar  to 


184  TELEMETERS.  [CHAP.  X 

rod  varies  as  y — the  distance  of  the  rod  from  the  outer 
focus  of  the  objective.  These  relations  may  be  seen 
directly  from  Fig.  54.  Since  the  two  rays  from/  and  q 
are  parallel  after  entering  the  telescope,  it  is  immaterial 
where  the  cross  hairs  are;  and,  therefore,  the  distance 
of  the  rod  from  e  is  always  proportional  to  the  intercept 
s.  In  other  words,  the  intercept  on  the  rod  is  propor- 
tional to  the  distance  of  the  rod  from  the  point  ^,  and 
any  change  in  the  position  of  the  cross  hairs  in  focusing 
upon  the  rod  produces  no  change  in  this  relation. 
From  Fig.  54  and  equation  (i)  we  get 

D=ks+c+f. (2) 

211.  As  equation  (2)  is  ihe  foundation  of  all  stadia 
formulas,  it  will  be  demonstrated  by  a  slightly  different 
method.  In  Fig.  55  o  represents  the  optical  centre  of 


/f\ 


-      C       — -», 


FIG. 


the  objective;  x  the  distance  from  the  cross  hairs  to 
the  optical  center  ;  and  z  the  distance  from  the  optical 
centre  to  the  rod.  The  remainder  of  the  nomenclature 
is  as  in  Fig.  54.  Since  a  lens  may  be  regarded  as  a 
mathematical  point  which  allows  a  very  great  amount 
of  light  to  pass  through  it,*  we  may  consider  the  rays 
of  light  as  passing  in  a  right  line  from  q  through  o  to  a, 
and  also  from  /  through  o  to  b.  Then  by  similar  tri- 
angles, 

s  •:  z  : :  i  :  x, 

*  See  first  foot-note  on  page  106. 


ART.    l]  THE    STADIA.  185 


from  which 


From  the  theory  of  optics,  we  have  the  well-known 
relation  for  a  convex  lens, 


Combining  equations  (3)  and  (4)  gives 

......     (5) 


Adding  c  to  both  members  of  (5)  gives  equation  (2), 
page  184. 

212.  There  are  other  and  less  simple  demonstrations 
of  the  fundamental  stadia  formula  —  equation  (2);  —  but 
as  they  all  arrive  at  the  same  final  form,  they  must  in- 
volve the  same  approximations.     Both  of  the  preceding 
methods  involve  slight  approximations:   i.  Equation  (i) 
assumes  that  the  line  qe,  Fig.  54,  and  the  horizontal  line 
through    a    intersect    in    a   vertical   plane   through   the 
optical  center,  whereas   they  do    not    so    intersect.     2. 
Equation  (3)  assumes  that  the  lines  qo  and  oa,  Fig.  55, 
are  one  and  the  same  right  line,  whereas  they  are  not. 

In  addition  to  these,  several  relatively  unimportant 
approximations  are  indirectly  involved.  Although  the 
formula  for  the  stadia  is  not  mathematically  correct, 
it  is  much  more  accurate  than  any  observations  that 
can  be  made  with  an  ordinary  telescope,  and  it  will  be 
shown  later  that  the  stadia  is  an  instrument  of  con- 
siderable precision. 

213.  To  find  c  and/.     To  find  c,  focus  the  instrument 
on  a  point   100  feet  or  more  away,  and  measure  with  a 


1 86  TELEMETERS.  [CHAP,  x 

pocket-rule  the  horizontal  distance  from  the  vertical 
axis  to  the  middle  of  the  objective.  Strictly,  c  is  not 
constant  in  instruments  in  which  the  cross  hairs  are 
fixed  and  the  objective  movable  ;  but  if  it  is  found 
within  an  inch  it  is  more  than  sufficient.*  The  distance 
found  as  above  is  practically  the  minimum,  but  it  is  also 
the  value  corresponding  nearly  to  the  mean  distance  of 
the  rod  from  the  instrument. 

To  find/,  focus  the  instrument  on  a  point  100  feet  or 
more  away,  and  measure  the  distance  from  the  cross 
hairs  to  the  middle  of  the  thickness  of  the  objective.! 

214.  To  find  k.  Set  the  rod  vertical,  at  any  convenient 
distance  in  front  of  the  instrument,  and  having  brought 
the  line  of  sight  horizontal,  determine  the  intercept  s 
by  using  a  stadia  rod  (§  207)  or  a  self-reading  level  rod 
(§  270).  Then  measure  the  distance  D  with  a  band- 
chain.  From  equation  (2),  page  184,  we  have 


*=-   -ir  -> <6) 

from  which  it  is  an  easy  matter  to  compute  k.  For 
greater  accuracy,  make  several  observations  at  different 
distances,  and  take  the  mean  of  the  corresponding 
values  of  k.  The  error  of  this  mean  will  be  consider- 


*  The  variation  in  c  is  the  same  as  the  variation  in  x  in  equation  (4), 
page  185.  Assuming/" to  be  i  foot — a  fair  average, — we  find  that  if  z  =  jo 
ft.,  x  —  i  ft.  1.3  in.;  if  z  =  20  ft.,  x  =  i  ft.  0.6  in.;  if  z  =  too  ft.,  x  =  i  ft. 
o.i  in.;  and  if  z  =  1000  ft.,  x  —  i  ft.  o.oi  in.  Therefore,  since  the  distance  to 
the  rod  will  ordinarily  be  more  than  100  ft.,  the  variation  in  c  is  less  than  one 
tenth  of  an  inch,  which  is  wholly  inappreciable  in  stadia  surveying. 

t  It  is  sometimes  desirable  to  know /"and  the  position  of  the  optical  center 
accurately.  To  find  them  proceed  as  follows  :  Remove  the  lens  from  the 
telescope  tube,  and  set  it  up  midway  between  two  small  movable  white  screens, 
one  of  which  has  a  square  ruled  upon  it  with  black  ink.  Shift  the  positions 
of  the  screens  until  the  square  and  its  image  upon  the  other  screen  are  of 
exactly  the  same  size.  The  optical  center  will  then  be  exactly  midway  between 
the  two  screens,  and  /is  equal  to  one  fourth  of  the  distance  between  them. 


ART.   ij  THE    STADIA.  187 

ably  less  than  the  error  of  a  single  determination  of  the 
intercept  s. 

The  above  method  may  be  employed  to  determine 
the  k  for  each  of  the  side  intervals.  If  the  stadia  hairs 
are  not  adjustable,  the  side  intervals  will  probably  not 
be  equal  to  each  other  ;  but  a  method  of  overcoming 
this  will  be  explained  later  (§  235,  second  paragraph). 
Of  course,  if  the  intervals  are  equal,  the  values  of  k  will 
be  equal  to  each  other  ;  and  in  any  case  the  sum  of  the 
reciprocals  of  k  for  the  side  intervals  must  be  equal  to 
the  reciprocal  of  k  for  the  extreme  interval,  which 
affords  an  excellent  check  on  the  accuracy  of  the  work. 

215.  For  convenience  of  computation,  k  is  ordinarily 
made  100  and  s  is  measured  in  feet.  This  has  the  fur- 
ther advantage  of  permitting  the  use  of  ari  ordinary 
level  rod  as  a  stadia  rod.  If  the  hairs  are  adjustable, 
this  condition  may  be  readily  attained.  To  adjust  the 
hairs,  carefully  draw  three  diamonds  with  black  ink  on 
cardboard,  and  fasten  two  of  these  targets  on  a  rod 
exactly  i  foot  apart,  and  place  the  third  one  midway 
between  the  other  two.  Then  from  the  plumb-line 
measure  a  distance  (100  -j-  ^-j-/)  feet  in  front  of  the 
instrument,  and  set  the  rod  vertical.  With  the  tangent 
screw  of  the  vertical  movement,  bring  the  central  hair 
to  bisect  the  middle  target,  and,  next,  with  the  stadia- 
hair  screws  bring  the  side  hairs  to  bisect  the  side 
targets  respectively.  Test  the  adjustment  by  placing 
the  outside  targets  2  feet  apart  and  setting  the  rod 
(200  -)-<:+/)  feet  from  the  instrument.  In  making 
these  observations,  the  line  of^sight  should  be  at  le^st 
nearly  horizontal  (see  §  219).  When  the  hairs  are  ad- 
justed as  above,  the  stadia  formula  becomes 

Z>ft.  =  100  jft.  +  (*-}-/)  ft.    .     .     .     (7) 


If  the  stadia  hairs  are  not  adjustable,  k  can  be  made 
equal  to   100  (or  any  other  number)  by  the  following 


I&8  TELEMETERS.  [CHAP    X 


method.  Measure  (loo-f-^H-/)  feet  in  front  of  the  in- 
strument and  set  the  rod  vertical.  Place  the  telescope 
horizontal  or  nearly  so,  fasten  a  target  on  the  rod  at 
about  the  height  of  the  point  covered  by  the  upper 
hair,  and  bring  this  hair  exactly  to  bisect  the  target. 
Then  have  an  assistant  place  the  other  target  so  that  it 
will  be  bisected  by  the  lower  stadia  hair.  If  now  the 
distance  between  the  two  targets  is  carefully  measured 
and  divided  into  one  hundred  equal  parts,  and  the 
graduation  is  continued  over  the  whole  length  of  the 
rod,  the  number  of  units  in  the  intercept  will  be  equal 
to  the  number  of  feet  the  rod  is  from  the  front  focus  of 
the  objective  ;  that  is  to  say,  for  this  rod  and  this  tele- 
scope, k  =  100. 

Since  it  would  probably  be  impossible  to  mark  each 
of  the  one  hundred  divisions  upon  the  rod  in  such  a  way 
as  to  make  them  visible  at  any  considerable  distance, 
it  is  best  to  mark,  say,  each  tenth  one  (see  Fig.  49, 
page  180)  and  estimate  the  units.  When  k  is  determined 
as  above,  the  stadia  formula  becomes 

D  ft.  =  10  R  +  (c+f)  ft,     ...     (8) 

in  which  R  is  the  number  of  figures  (spots)  included  in 
the  intercept. 

216.  It  is  possible  to  determine  both  k  and  (c  -f-/)  by 
the  method  explained  in  §  214.  Make  two  observations 
for  s  at  two  known  distances,  Z>,  and  insert  the  results 
in  equation  (2),  page  184,  which  gives  two  equations  with 
two  unknown  quantities,  from  which  k  and  (c  -\-  f)  can 
be  determined  by  the  ordinary  rules  of  algebra.  For 
greater  accuracy  make  several  observations,  each  two 
of  which  will  give  a  value  of  each  of  the  unknown  quan- 
tities. Strictly  speaking,  the  resulting  equations  should 
be  solved  by  the  method  of  least  squares  ;  but  as  the 
method  of  finding  c,  /,  and  k,  as  explained  above,  is  very 


ART.   l]  THE    STADIA. 


much  more  simple  and  is  abundantly  exact,  this  method 
will  not  be  discussed  further,  even  though  it  is  practiced 
by  some  prominent  engineers. 

Frequently  in  finding  k,  the  intercept  is  assumed  to 
vary  as  the  distance  from  the  center  of  the  instrument, 
which  is  equivalent  to  omitting  (c-\-f)  from  equa- 
tion (6),  page  186.  In  this  case,  the  distance  as  deter- 
mined by  the  stadia  is  always  in  error,  except  when 
it  is  the  same  as  the  distance  employed  in  finding  k. 
For  shorter  distances  the  result  is  too  small,  and  for 
greater  distances  it  is  too  large.  Sometimes  observa- 
tions are  made  at  several  distances  and  a  mean  value 
for  k  is  computed,  which  simply  averages  errors  and 
makes  it  impossible  to  find  either  the  amount  or  direc- 
tion of  the  error.  This  method  is  incorrect  in  principle, 
and  inaccurate  in  practice,  and  has  nothing  to  commend 
it.  The  method  of  §  215  is  simple  and  gives  strictly 
correct  results  ;  and  if  approximate  results  are  desired 
the  quantity  (c -\- f)  in  equation  (2)  may  be  disregarded 
in  computing  the  distance,  in  other  words,  this  quan- 
tity may  be  omitted  or  inserted  at  will  without  intro- 
ducing error  into  any  other  part  of  the  work. 

217.  To  explain  a  method  of  eliminating  (c  +/)  from 
the  formula,  conceive  that  the  rod  has  been  graduated 
for  the  formula  D  =  k  s -\-  (c-}-f)  and  that  it  is  standing 
(ioo-f-^+/)  feet  from  the  instrument.  The  intercept 
may  be  considered  as  divided  into  100  divisions,  each  of 
which  corresponds  to  a  foot  of  distance  on  the  ground. 
Then  if  the  rod  is  shortened  by  cutting  out  of  the  in- 
tercept as  many  divisions,  or  units  of  the  graduation, 
as  there  are  feet  in  the  distance  (c  +/),  the  numbering 
of  the  graduations  remaining  unchanged,  the  apparent 
number  of  units  between  the  stadia  hairs  will  corre- 
spond to  the  number  of  feet  from  the  rod  to  the  center  of 
the  instrument.  If  care  is  taken  that  the  point  at  which 
a  portion  of  the  graduation  is  cut  out  is  always  between 


190  TELEMETERS.  [CHAP,  x 

the  stadia  hairs,  the  reading  will  give  the  distance  from 
the  center  of  the  instrument.  This  method  is  objec- 
tionable in  that  the  point  at  which  the  graduation  is 
omitted  can  not  always  be  made  to  fall  between  the 
stadia  hairs,  owing  to  obstructions  to  sighting  ;  and, 
further,  in  that  the  determination  of  the  vertical  co- 
ordinate is  complicated  thereby.  The  advantage  of 
having  the  intercept  proportional  to  the  distance  is  so 
slight  that  it  does  not  seem  wise  to  obtain  it  by  com- 
plicating both  the  graduation  and  the  use  of  the  rod. 
Furthermore,  the  stadia  is  not  intended  for  an  instru- 
ment of  high  precision,  and  the  work  for  which  it  is 
generally  used  does  not  require  extreme  accuracy  ; 
therefore,  as  a  rule,  the  term  (c  +/)  in  the  final  formula 
may  simply  be  omitted. 

Parro,  in  1823,  showed  that  by  placing  an  auxiliary 
lens  between  the  objective  and  the  cross  hairs,  the  in- 
tercept can  be  made  proportional  to  the  distance  from 
the  center  of  the  instrument.  It  is  not  known  that  such 
a  telescope  has  ever  been  made. 

218.  POSITION  OF  ROD  FOR  INCLINED  LINE  OF  SIGHT. 

The  preceding  formulas  were  deduced  on  the  assump- 
tion that  the  central  visual  ray  was  horizontal  and  tke 
rod  vertical,  i.e.,  the  central  visual  ray  was  assumed  to 
be  perpendicular  to  the  rod.  These  formulas  would  be 
sufficient  if  the  observations  were  made  with  a  leveling 
instrument;  but  a  level  is  too  limited  in  its  range  to 
secure  the  full  advantage  of  the  principle  of  the  stadia. 
In  all  that  follows,  it  will  be  assumed  that  a  transit  is 
used. 

The  formula  D  =  k s  -f-  c  -\- f  may  be  used  with  an 
inclined  line  of  sight,  provided  the  rod  is  held  perpen- 
dicular to  the  central  visual  ray.  In  this  case  D  is  no 
longer  the  horizontal  distance  from  the  instrument  to 
the  rod,  but  is  the  oblique  distance  from  the  horizontal 
axis  of  the  telescope  to  the  point  on  the  rod  covered 


ART.    l]  THE    STADIA. 


by  the  central  visual  ray.  It  then  becomes  a  question 
whether,  with  an  inclined  line  of  sight,  the  rod  should 
be  held  perpendicular  to  the  central  visual  ray,  or  verti- 
cal. 

The  perpendicularity  of  the  rod  to  the  line  of  sight 
may  be  determined  by  a  telescope  or  a  pair  of  sights 
attached  at  right  angles  to  the  rod,  which  is  directed 
toward  the  observing  telescope  by  the  rod-man.  The 
verticality  of  the  rod  may  be  estimated  by  the  rod-man, 
or  it  may  be  determined  easily  and  accurately  by  at- 
taching a  plumb-line  or  a  level  vial. 

Some  prefer  the  rod  perpendicular  to  the  line  of 
sight,  but  this  position  involves  serious  difficulties:  (i) 
it  is  not  easy  to  hold  the  rod  steady  in  this  position; 
(2)  it  is  not  always  possible  for  the  rod-man  to  see  the 
telescope,  especially  at  long  distances  or  great  vertical 
angles,  or  when  undergrowth  intervenes;  and  (3)  the 
formulas  for  computing  the  horizontal  and  vertical  co- 
ordinates of  the  point  are  more  simple  when  the  rod  is 
vertical  than  when  it  is  perpendicular  to  the  line  of 
sight.  Only  the  case  of  the  vertical  rod  will  be  con- 
sidered here. 

219.  FORMULAS  FOE  INCLINED  LINE  OF  SIGHT  AND  VER- 
TICAL ROD.  Let  0  —  the  angle  of  the  central  visual  ray 
with  the  horizontal  =  C//(Fig.  56).  To  measure  8,  a 
third  horizontal  hair  should  be  placed  half-way  between 
the  other  two,  or,  rather,  stadia  hairs  are  to  be  added  on 
opposite  sides  and  equally  distant  from  the  ordinary 
horizontal  one;  and  6  is  then  determined  as  any  other 
vertical  angle  by  reading  the  vertical  circle.  8  will 
generally  be  small.  Let  2tx  =  the  visual  angle  —  BOD. 
2a  is  always  small,  its  maximum  being  about  35  min- 
utes. AE  is  the  actual  intercept,  and  BD  the  value  it 
would  have  if  the  rod  were  held  perpendicular  to  the 
central  visual  ray.  It  is  desired  to  find  a  relation  be- 
tween AE  and  BD, 


I92 


TELEMETERS. 


[CHAP.  X 


The  angle  CBA  =  90°  -j-  a\  and,  since  a  is  very  small, 
BC —  AC  cos  6  nearly.  Similarly  CDE  —  90°  —  a,  and 
CD=CE  cos  0  nearly.  Hence  BD-(AC-\-CE}  cos  8  = 
AE  cos  6  nearly.  Notice  that  the  two  approximations 


FIG.  56. 

tend  to  neutralize  each  other,  and  that  the  final  error 
involved  is  much  less  than  the  error  of  observing  AE* 

Since  BD  is  the  intercept  perpendicular  to  the  line  of 
sight,  it  is  the  value  of  s  (§  210)  corresponding  to  the 
distance  1C. 

Hence  equation  (2),  page  184,  becomes 

D  =  k  s  cos  6  -\-  (c  -4-  /),  (o) 

I      \         I     J   />  \s/ 

in  which  D  is  the  oblique  distance  1C  (Fig.  56). 

220.  The  Horizontal  Distance.  Let  H—  the  horizontal 
distance  from  the  center  of  the  instrument  to  the  verti- 
cal through  the  foot  of  the  rod  =  //  (Fig.  56).  H  —  1C 
cos  9  —  D  cos  6;  and  substituting  the  value  of  D  from 
equation  (9),  we  have 

0+(c+f)cosB,       .;•;.'.     (10) 


=s  cos 


or 


J?=  ks  —  ks  sin*  0+  (c+f)  cos  0. 


*  If  the  side  hairs  are  equidistant  from  the  central  one,  the  true  relation  is 
BD  =  AE  cos  Q(i  —  tan?  Q  tan?  a) ;  and  if  the  side  hairs  are  not  equidis- 
tant, the  upper  angle  being  a'  and  the  lower  a,  ,  the  true  relation  is 


ART.   l]  THE    STADIA.  193 

The  second  form  is  preferable,  for  it  is  always  better 
to  compute  a  correction  to  a  quantity  than  the  quantity 
itself.  Notice  that  since  (c  +  f)  is  always  small  and 
cos  9  nearly  unity,  (c  -J- /)  cos  6  may  always  be  taken 
equal  to  (^+/),  and  often  may  be  omitted  entirely. 
Notice  also  that  since  si/i*  B  is  small,  the  whole  opera- 
tion of  reducing  an  observation  by  equation  (n)  may 
be  performed  mentally. 

221.  Vertical  Distance.     To  determine  the  vertical  co- 
ordinate of  the  point  upon  which   the  rod  is  set,  the 
middle  hair  must  be  sighted  upon  a  point  of  the  rod  at 
a  known  distance  from  its  foot.     The  simplest  way  of 
accomplishing  this  is  to  provide  the  rod  with  a  movable 
target  (§  209). 

If  the  instrument  is  set  over  the  reference  point,  the  target 
is  to  be  set  at  a  distance  from  the  foot  of  the  rod  equal 
to  the  height  of  the  horizontal  axis  of  the  telescope 
above  the  point.  The  proper  position  of  the  target 
may  readily  be  found  by  setting  the  rod  up  by  the  side 
of  the  instrument.  Then  the  difference  in  level  be- 
tween the  point  under  the  instrument  and  the  one  on 
which  the  rod  is  placed  is  equal  to  the  diffevence  in  ele- 
vation between  the  horizontal  axis  of  the  telescope  and 
the  target  on  the  rod;  and  therefore  to  determine  the 
height  of  any  subsequent  point,  set  the  rod  upon  it, 
bisect  the  target,  and  read  the  angle  from  the  vertical 
circle. 

If  the  instrument  is  not  placed  over  the  reference  point, 
bring  the  line  of  sight  horizontal,  set  the  rod  on  the 
reference  point,  and  move  the  target  until  it  is  bisected 
by  the  middle  cross  hair.  To  determine  the  height  of 
any  subsequent  point,  set  the  rod  upon  it  and  read  ex- 
actly as  before. 

222.  To  deduce  a  formula  for  computing  the  vertical 
co-ordinate,  let   V  =  the  height  of  the  point  on  which 
the  rod  is  placed,  above  the  reference  point;    F~the 


194  TELEMETERS.  [CHAP,  X 

distance  the  target  is  above  the  axis  of  the  telescope  = 
JC  (Fig.  56).  V—  1C  sin  6  =  D  sin  0;  and  substitut- 
ing the  value  of  D  from  equation  (9)  gives 

V  =  k  s  cos  0  sin  6  +  (c  +/)  sin  0;   .     .     (12) 
V  =  \ks  sin  26+  (c+/)  sin  6.       .     .     (13) 

Notice  that  generally  (c  +/)  sin  6  may  be  omitted. 

223.  REDUCING  THE  FIELD  NOTES.  This  consists  in 
finding  the  horizontal  and  vertical  distances  from  the 
rod  reading  and  the  observed  vertical  angle.  This  can 
be  done  by  using  formulas  (10)  and  (13).  Although  the 
equations  are  in  a  very  convenient  form  for  computa- 
tion, it  would  be  very  tedious  and  slow  to  solve  both 
equations  for  every  observation. 

Notice  that  ks  cos*  0  and  \ks  sin  2  8  are  independent 
of  the  instrument  and  of  the  unit  of  linear  measurement 
used,  and  therefore  they  can  be  tabulated  for  all  cases. 
(c  -{-  f)  cos  9  and  (c  -f-  f)  sin  8  depend  upon  both  the  in- 
strument and  the  unit,  and  must  be  computed  once  for 
each  particular  instrument  and  rod.  Obviously  it  is  a 
great  saving  of  labor  and  time  if  the  results  are  com- 
puted once  for  all  and  tabulated. 

We  may  compute  ks  cos*  8  from  equation  (10),  and 
\ks  sin  2  8  from  equation  (13),  for  different  values  of  ks 
and  0,  and  tabulate  the  results,  in  which  case  the  hori- 
zontal and  vertical  distances  can  be  taken  directly  from 
the  table ;  or,  we  may  tabulate  only  cos*  8  and  \  sin  2  8 
for  different  values  of  0,  in  which  case  the  horizontal 
and  vertical  distances  are  found  by  multiplying  ks  by 
the  tabulated  factor.  The  first  method  would  be  the 
better,  if  it  did  not  require  such  voluminous  tables. 

In  either  case  the  results  may  be  expressed  in  an 
arithmetical  table  or  in  a  geometrical  diagram.  Arith- 
metical tables  are  capable  of  greater  accuracy;  but  they 


ART.   l]  THE   STADIA.  195 

must  be  either  very  extended  and  therefore  inconveni- 
ently large,  or  brief  and  give  results  by  interpolation, 
which  is  slow  and  tedious.  On  the  other  hand,  it  is 
urged  against  geometrical  diagrams  that  to  be  accurate 
they  must  be  drawn  to  a  large  scale,  and  that  therefore 
they  are  large  and  unwieldy.  It  is  believed  that  with 
properly  constructed  diagrams  the  reductions  can  be 
made  without  sacrificing  much,  if  any,  accuracy,  and 
with  greater  facility  than  by  the  use  of  tables. 

224.  Arithmetical  Tables.     Pages  196  to  198  contain  a 
brief  arithmetical  reduction  table.     The  column  headed 
H  gives  the  value  of  sin9  6  (see  equation  (n),  page  192). 
The  column  headed  V  gives  the  value  of  \  sin  2  6  (see 
equation   (13),  page  194).     The  terms  (c  +/)  cos  0  and 
(c  -f-jO  sin  tt  are  seldom  required ;  but  they  are  given  in 
a  note  at  the  bottom  of  the  page,  for  use  when  great  ac- 
curacy is  desired. 

Ockerson  and  Teeple,  assistant  U.  S.  engineers,  have 
published  tables  *  which  give  \  k  s  sin  2  0  -f~  0.43  in  metres 
(for  values  of  ks  in  feet)  from  o  to  500,  varying  by  10, 
for  each  minute  of  Q  from  o°  to  10°.  These  tables  give 
also  sin*  B  -f  0.43  for  the  even  minutes  from  2°  to  20°. 

225.  Geomatrical  Diagrams.      There   is   a  variety   of 
methods  of  constructing  diagrams  for  reducing  stadia 
observations,  but  a  modification  of  those  proposed  by 
Prof.  S.  W.  Robinson  f  are  believed  to  be  the  best. 

226.  Horizontal  Distance.      Fig.    57   (between    pages 
200  and  201)  gives  the  term^j  sin*  0  (see  equation  (n), 
page  192).     To  use  the  diagram,  find  the  observed  value 
of  ks  on  the   lower   line,  and   from    this  point  follow 
the    inclined    line    to    the    radial    line    indicating   the 
observed  value  of  6  ;  then  the  elevation  of  this  point  as 

*  For  sale  by  J.  A.  Ockerson,  Chief  Engineer  Mississippi  River  Commis- 
sion, St.  Louis,  Missouri. 

t  Journal  of  the  Franklin  Institute,  Vol.  49,  pp.  80,  81 ;  and  Ibid.,  Vol, 
51,  pp.  15,  16. 


196 


TELEMETERS. 


[CHAP.  X 


TABLE   III. 
STADIA  REDUCTION  TABLES. 


9 

o' 

2 

o° 

1° 

2 

3° 

4° 

5° 

H 

v. 

H 

v 

H 

v. 

H 

v. 

H. 

v. 

H. 

v 

.0027 
.0028 

•0523 
.0528 

.0049 
.0049 

.0696 

.0702 

.0076 
.0077 

.0868 

.0874 

.0000 
.0000 

.0000 

.0006 

.0003 
.0003 

.0174 
.0180 

.OOIK 

.0013 

•°349 

•0355 

I 

.0000 
.0000 

.0012 

.0017 

.0003 
.0004 

.0186 

.0192 

.0013 
0013 

.0360 

.0366 

.0029  .0534 
.0029  .0540 

.0050 
.0051 

.0707 

•°7*3 

.0078 
.0079 

.0880 
.0885 

8 
10 

.0000 
.0000 

.0023 
.0029 

.0004 
.0004 

.0197 

.0203 

.0014 
.0014 

.0372 
.0378 

.0030 
.0030 

.0546 
.0552 

.0052 

•0053 

.0719 
.0725 

.0080 
.0081 

.0801 

.0897 

12 
*4 

.0000 
.0000 

.0035 

.0041 

.0004 
.0004 

.0209 
.0215 

.0015 
.0015 

.0384 
.0390 

.0031 
.0032 

•P5S7 

•0563 

.0054 
.0054 

.0730 
.0736 

.0082 
.0083 

.0903 
.0908 

16 
18 

.0000 
.0000 

.0046 

.0052 

.0005 
.0005 

.0221 
.0227 

.0016 

.0016 

•0395 

.0401 

.0032 
.0033 

.0569 
•0575 

.0055 
.0056 

.0742 
.0748 

.0084 
.0085 

.0914 
.0920 

20 
22 

.0000 

.0000 

0058 

.0064 

.0005 

.0005 

.0232 

.0238 

.0017 
.0017 

.0407 
.0413 

.0034 
.0035 

.0580 

.0586 

.0057 
.0058 

•0753 

.0759 

.0086 
.0087 

.0925 
.0931 

24 
26 

.0000 

.0000 

.0070 

.0076 

.0006 
.0006 

.0244 
.0250 

.0017 
.0018 

.0418 
.0424 

•  0035 
.0036 

.0592 
.0598 

.0059 

.0060 

.0765 

.0771 

.0088 
.0089 

•0937 
•0943 

28 
30 

.0001 
.0001 

.0081 
.0087 

.0007 

.0007 

.0256 
.0261 

.0018 

.0019 

.0430 
.0436 

.0037 
.0037 

.0604 
.0609 

.0061 
.0062 

.0776 
.0782 

.0090 
.0091 

.0948 

•OQ54 

32 

34 

.0001 

.0001 

.0093 
.0099 

.0007 

.0008 

.0267 

.0273 

.0019 
.0020 

.0442 
.0448 

.0038 
.0039 

.0615 
.0621 

.0062 
.0063 

.0788 
.0794 

.0092 
.0093 

.0960 
.0965 

g 

.0001 

.0001 

.0105 

.OIIO 

.0008 
.0008 

.0279 

.0285 

.0021 

.0021 

•0453 
•0459 

.0039 

.0040 

.0627 

•0633 

.0064 
.0065 

.0799 

.0805 

.0094 
•  0095 

.0971 
.0977 

40 
42 

.0001 

.0002 

.0116 
.0122 

.0009 
.0009 

.0291 
.0297 

.0022 

.0022 

.0465 
.0471 

.0041 
.0042 

.0638 
.0644! 

.0066 

.0067 

.0811 
.0817 

.0097 
.0098 

.0983 
.0988 

8 

48 
5° 

.0002 

.0002 

.0002 
.0002 

.0128 
.0134 

.0139 

.0145 

.0009 

.0010 
.0010 

.0010 

.0302 
.0308 

.0314 
.0320 

.0023 
.0023 

.0024 
.0024 

.0476 
.0482 

.0488 
.0494 

.0042 
.0043 

.0044 
.0045 

.0650 

.0656 

.0661 
.0667 

.0068 
.0069 

.0070 
.0071 

.0822 
.0828 

.0834 

.0840 

.0100 
.OIOI 

.0102 
.0101 

.0994 

.  IOOO 

.  1005 
ion 

52 

54 

.0003 
.0003 

.0151 

•0157 

.0011 
.OOII 

.0326 
•Q331 

.0025 
.0026 

.0499 
.0505 

.0045 

.0046 

.0673 
.0678 

.0072 
.0073 

.0845 
.0851 

.0104 
.0105 

.  1017 

.1022 

56 
58 

.0003 

.0003 

.016} 

.0168 

.001  1 
.oon 

•0337 
•0343 

.0026 
.0027 

.0511 

•0517 

.0047 
.0048 

.0684 

.0690 

.0074 
•  0075 

.0857 

.0863 

.0107 
.0108 

.1028 

.1034 

Co 

.0003 

•°*74 

.0012 

.0349 

.0027 

.0522 

.0049 

.0696 

.0076 

.0868 

.OIOC 

.1040 

For  all  /alues  of  Q  found  on  this  page 


(c  +/)«'»  9  =  0. 


ART.   l] 


THE    STADIA. 


I97 


TABLE   III. 

STADIA  REDUCTION  TABLES. 


0 

6° 

7° 

8° 

9° 

10° 

11° 

H. 

V. 

H. 

V. 

H. 

V. 

H. 

.0245 
.0247 

V. 

•  1545 
•J551 

H. 

.0302 
.0304 

V. 

.1710 

.  1716 

H. 

•0364 
.0366 

V. 

•  1873 

.1878 

o 

2 

.0109 

.0110 

.  1040 
.1045 

.0149 
.0150 

.1210 
.1215 

.0194 
.0195 

.1378 
.1384 

\ 

.0112 

.0113 

•  1051 
•  1057 

0151 
•  0153 

.1221 
.1226 

.0197 
.0199 

•  1389 
.1395 

0248 
.0250 

.1556 

.1562 

.0306 
.0308 

.1721 
.1726 

.0368 
•0371 

.1884 
.1889 

8 

10 

.0114 
.0115 

.1062 
.1068 

•0154 
.0156 

.1232 

.1238 

.0200 
.0202 

.1401 
.  1406 

.0252 
.0254 

•1567 
•I573 

.0310 
.0312 

•  1732 
•I737 

•0373 
•°375 

.1895 

.1900 

12 

14 

.0117 

.0118 

.1074 
.1079 

•OI57 
.0158 

.1243 
.1249 

.0203 
.0205 

.1412 
.1417 

.0256 
.0257 

•1578 
.1584 

.0314 
.0316 

•1743 
.1748 

•°377 
°379 

•  1905 
.1911 

16 
18 

.0119 

.0120 

-1085 
.10911 

.0160 
.0161 

.1255 
.  I26o 

.0207 
.0208 

•  1423 
.1428 

.0259 
.0261 

.1589 
•1595 

.0318 
.0320 

•1754 
•1759 

.0382 
.0384 

.1916 
.1921 

20 
22 

.OI22 
.0123 

,0,6 

.  IIO2 

.0163 
.0164 

.1266 
.1272 

.O2IO 
.0212 

•  1434 
.1440 

.0263 
.0265 

.1600 
.1606 

.0322 
.0324 

-1765 
.1770 

.0386 
.0388 

.1927 
.1932 

24 
26 

.OI24 
.0126 

.  1  108 
•"'3 

.0166 
.0167 

.1277 
.1283 

.0213 
.0215 

•  1445 
•  1451 

.0267 
.0269 

.1611 
.1617 

.0326 
.0328 

.1776 
.1781 

.0391 
•0393 

.1938 
•  1943 

28 

3° 

.0127 
.OI28 

.1119 
•"25 

.0169 
.0170 

.1288 
.1294 

.O2I7 
.02l8 

•  1456 

.  1462 

.0271 

.0272 

.  1622 
.1628 

-0330 
•  0332 

.1786 
.1792 

•0395 
•°397 

.1948 
•  1954 

32 
34 

.0129 
.0131 

•  II3°] 
.1136 

.0172 
•0173 

.I300 
•1305 

.O22O 
.0222 

.1467 
•  1473 

.0274 
.0276 

.1633 
.1639 

•0334 
•  0336 

.1797 
.1803 

.0400 
.0402 

•  1959 
.1964 

* 

.0132 
.0133 

.1142 
.1147 

•0175 
.0176 

.13" 
-1317 

.0224 
.O225 

•*479 
.1484 

.0278 
.0280 

.1644 
.  1650 

•  0338 
.0340 

.1808 
.1814 

.0404 
.0407 

.1970 
•  1975 

40 
42 

•0135 
.0136 

•"53 
•"59 

.0178 
.0180 

.1322 
.1328 

.0227 
.0229 

.1490 
•H95 

.0282 
.0284 

•1655 
.1661 

.0342 
,  -0345 

.1819 
.1824 

.0409 
1.0411 

.1980 
.1986 

44 
46 

•0137 
.0139 

.  1164 
.1170 

.0181 
.0183 

•1333 
•1339 

.0231 
.0232 

.1501 
.1506 

.0286 
.0288 

.1666 
.1672 

•°347 
•0349 

.1830 
•1835 

.0414 
.0416 

.1991 
.1996 

48 

So 

.0140 
.0142 

.1176 
.1181 

.0184 
.0186 

•I345 
•135° 

.0234 
.0236 

.1512 
•I5I7 

.0290 
.0292 

.1677 
.1683 

•0351 
•0353 

.1841 
.1846 

.0418 
.0421 

.2002 
.2007 

52 
54 

.0143 
.0144 

.1187 
•"93 

0187 
.0189 

:$ 

.0238 
.0239 

•!523 

.1528 

.0294 
.0296 

.1688 
.1694 

•°355 
.0358 

.1851 
.1857 

.0423 
.0425 

.2012 

.20l8 

£ 

60 

.0146 
.0147 

.0149 

.1198 

.1204 

.M.o| 

.0190 
.0192 

.0194 

•1367 
•1373 

•1378 

.0241 
.0243 

.0245 

•1534 
.1540 

•1545 

.0298 
.0300 

.0302 

.1699 
•I7°5 

.1710 

.0360 
.0362 

.0364 

.1862 
.1868 

•1873 

.0428 
.0430 

.0432 

.2O23 
.2028 

.2034 

For  values  of  6  on  this  page  -j 


sin  0  varies  between  o.i(c-\-f)  and  o.z(c-}-/^ 


198 


TELEMETERS. 


[CHAP,  x 


TABLE  III. 
STADIA  REDUCTION  TABLES. 


0 

I28 

13° 

14° 

'5° 

it 
H. 

.0760 
.0763 

V. 

2650 
•2655 

17° 

H. 

V. 

H. 

V. 

H. 

V. 

H. 

V. 

2500 
.2505 

H. 

V. 

o' 

2 

.0432 
•°435 

.2034 
.2039 

.0506 
.0509 

.2192 
.2197 

.0585 
.0588 

•2347 
•2352 

.0670 
.0673 

0855 
0858 

2796 
2801 

4 
6 

•°437 
•°439 

.2044 

.2050 

.0511 
•<>5I3 

.2202 
.2208 

.0591 
•0594 

.2358 
•  2363 

.0676 
.0679 

2510 
•2515 

.0766 
.0769 

.2659 
.2664 

0861 
0865 

2806 

2810 

8 
10 

.0442 
.0444 

•2055 

.2060 

.0516 
.0519 

.2213 
.2218 

.0596 
•0599 

.2368 
•2373 

.0682 
0685 

.2520 
•2525 

.0772 
•0775 

.2669 
.2674 

0868 
0871 

2815 
2820 

12 

M 

.0447 
.0449 

.2066 
.2071 

.0521 
.0524 

.2223 
.2228 

.0602 
.0605 

.2378 
•2383 

.0687 
.0690 

•2530 
•2535, 

.0778 
.0782 

.2679 
.2684 

0874 
0878 

2825 
2830 

16 

18 

•  Q451 
•°454 

.2076 

.2081 

.0527 
.0529 

.2234 
.2239 

.0607 
.0610 

.2388 
•2393 

.0693 
.0696 

.2540 
•2545 

.0785 
.0788 

.2689 
.2694 

0881 
0884 

.2834 

.2830 

20 
22 

.0456 
•0459 

.2087 
.2092 

.0532 
•0534 

.2244 
.2249 

.0613 
.0010 

•2399 
.2404 

.0699 
.0702 

•255° 

•2555 

.0791 
.0794 

.2699 
.2704 

.0888 
.0891 

.2844 
.2849 

24 
26 

.0461 
.0464 

.2097 

.2103 

•0537 
.0540 

.2254 
.2260 

.0619 
.0621 

.2409 
.2414 

.0705 
.0708 

.2560 
•2565 

.0797 
.0800 

.2709 
.2713 

.0894 
.0898 

2854 
.2858 

28 
30 

.0466 
.0469 

.2108 

.2113 

.0542 
•0545 

.2265 
.2270 

.0624 
.0627 

.2419 
.2424 

.0711 
.0714 

.2570 
•2575 

.0804 
.0807 

.2718 
•  2723 

0901 
,0904 

.2863 
.2868 

32 
34 

IS 

.0471 
•0473 

0476 
.0478 

.2118 
.2124 

.2129 
.2134 

.0548 
•0550 

•0553 
•0556 

.2275 
.2280 

.2285 
.2291 

.0630 
•0633 

.2429 
•2434 

•  2439 
.2444 

.0717 
.0720 

.0723 
.0726 

.2580 
•  2585 

.2590 
•2595 

.0810 
•0813 

.0816 
.0819 

.2728 
•2733 

•2738 
•2743 

.0908 
.0911 

°9'4 
.0910 

.2873 
.2877 

.2882 
.2887 

40 

42 

.0481 
.0483 

.2139 
•2145 

•0558 
•  0561 

.2296 
.2301 

.0641 
.0644 

.2449 
•  2455 

.0729 
.0732 

.2600 
.2605 

.0822 
.0826 

-2748 
.2752 

.0921 
.0924 

.2892 

.2896 

44 
46 

.0486 
.0488 

.2150 
•  2155 

.0564 
.0566 

.2306 
.2311 

.0647 
.0650 

.2460 
.2465 

.0735 
.0738 

.2610 
.2615 

.0829 
.0832 

•2757 
.2762 

.0928 
.0931 

.2901 
.2906 

48 
5° 

.0491 
•0493 

.2160 
.2166 

.0569 
.0572 

.2316 
.2322 

.0653 
.0656 

.2470 
•2475 

.0741 
.0744 

.2620 
.2625 

.0835 
.0839 

.2767 
.2772 

.0935 
.0938 

.2911 
.2915 

52 
54 

.0496 
.0498 

.2171 

.2176 

•°575 
•0577 

.2327 
.2332 

.0658 
.0661 

.2480 
.2485 

.0747 
.0750 

.2630 
.2635 

.0842 
.0845 

.2777 
.2781 

.0941 
•0945 

.2920 
.2925 

s 

60 

.0501 
.0504 

.0506 

.2181 
.2187 

.2192 

.0580 
.0583 

•0585 

•2337 

•*34° 

•2347 

.0664 
0667 

.0670 

.2490 
•2495 

.2500 

•0754 
•0757 

.0760 

.2640 
.2645 

.2650 

.0848 
0852 

•0855 

.2786 
.2791 

.2796 

.0948 
.0952 

•°9S5 

.2930 
•2934 

•  2939 

For  valuesjof  Q  on  this  page 


i  c+f)  cos  0  >  o.9s(tf +/)  = 

\  (c  +/")  sin  fl  varies  between  o.2(c  -f /)  and  0.3(4:  -j-/> 


ART.  l] 


THE    STADIA. 


I99 


TABLE    III. 
STADIA  REDUCTION  TABLES. 


9 

o' 

2 

iS 
H. 

JO 

V. 

19° 

20° 

21° 

22° 

23- 

H. 

V. 

H. 

V. 

H. 

V. 

H. 

.1403 
.1407 

V. 

3473 
3477 

H. 

V. 

3597 
3601 

•°95S 
.0958 

•2939 
•2944 

.  1060 
.1064 

•3078, 
•3083 

.1170 
•"74 

.3214 

.3218 

.1284 
.1288 

3346 
3350 

•1527 
•i53i 

46 

.0962 
.0965 

.2948 
•  2953 

.1067 
.  1071 

.3087 
.3092 

•"77 
.1181 

.3223 
•3227 

.1292 
.1296 

3354 
3359 

.1411 
.1416 

3482 
3486 

•1535 
•1539 

3605 
3609 

•  8 

10 

.0969 
.0972 

.2g=;8 
.2962 

.1074 
.1078 

•3097 
.3101 

•  "85 
.1189 

.32321 

.3236 

.1300 
.1304 

3363 
3367 

.1420 
.1424 

'490 
3494 

3* 

3613 
36i7 

12 
Z4 

.0976 
.0979 

.2967 
.2972 

.1082 
.1085 

.3106 
.3110 

.  1192 
.1196 

.3241 
•3245 

•  1308 
.1312 

3372 
3376 

.1428 
•M32 

3498 
3502 

•  1552 
•  1556 

3621 
3625 

16 
18 

.0982 
.0986 

.2976 
.298. 

.1089 
.1092 

•3"5 
•3"9 

.1200 
.1204 

•3249 
•3254 

•  1316 
.1320 

338o 
3384 

.1436 
.1440 

3507 
35" 

.1560 
•  1565 

3629 
3633 

20 

92  j 

.0989 
•0993 

.2986 
.2990 

.1096 

.1099 

•3124 

.3128 

.  I2O7 
.1211 

.3258 

•  3263 

•1323 
•I327 

3389 
3393  ! 

.1444 
.1448 

3515 
3519 

.1569 
•  1573 

3637 
3641 

3 

.0996 

.  IOOO 

•2995 
.3000 

.1103 
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20O  TELEMETERS.  [CHAP.  X 

given  by  the  scale  at  the  left  of  the  diagram  is  the 
value  of  ks  sin*  6,  which  is  to  be  subtracted  from  ks. 
For  extreme  accuracy  add  (<:+/).  For  example,  if  ks 
=  740  and  8  =  10°,  what  is  JfJ  On  the  bottom  line  find 
740,  and  follow  the  inclined  line  to  an  intersection  with 
the  radial  line  marked  10°.  By  the  scale  on  the  left 
this  point  reads  22.4.  Therefore  Z>  =:  740  —  22.44- 


If  6  is  more  than  13°,  the  method  of  using  the  dia- 
gram differs  slightly  from  that  just  described.  For 
example,  if  ks  =  820  and  6  —  17°,  what  is  HI  On  th| 
extreme  right-hand  line  of  the  diagram,  find  820  and 
follow  this  horizontal  line  to  its  intersection  with  the 
radial  line  marked  17°,  and  then  follow  this  oblique 
line  to  the  scale  at  the  top,  we  see  the  value  sought  is 
70.  Therefore  H=  820  —  70  +  (c+f). 

227.  Vertical  Distance.  Fig.  58  (between  pages  200 
and  201)  gives  the  ralue  of  ^kssin2  6  (see  equation 
(13),  page  194). 

This  diagram  is  used  in  essentially  the  same  manner 
as  Fig.  57  (§  226).  If  the  observed  value  of  9  is  found 
in  the  left-hand  triangle  or  section  of  the  diagram,  the 
value  of  ks  is  to  be  found  on  the  bottom  line  to  the  left 
of  the  center,  and  the  final  result  is  found  on  the  left 
edge  of  the  left-hand  triangle.  If  the  value  of  6  is 
found  in  the  middle  section,  the  value  of  ks  may  be 
found  as  before,  then  following  the  inclined  line  up  to 
the  6°  line,  and  from  this  point  following  a  horizontal 
line  to  the  radial  line  corresponding  to  0,  and  from 
thence  following  the  oblique  line  to  the  scale  on  the 
upper  edge  of  the  drawing,  find  the  value  sought  ;  or  if 
preferred,  the  value  of  k  s  and  also  of  the  final  result  may 

e  found  by  the  numbers  written  upon  the  face  of  the 
diagram.  If  the  given  value  of  0  is  found  in  the  right- 
hand  triangle  or  section  of  the  diagram,  find  the  value 
of  k  s  on  the  bottom  line  to  the  right  ot  the  center,  and 


ART.    l]  THE    STADIA.  2OI 

follow  the  more  inclined  lines  to  the  intersection  with 
the  proper  radial  line  ;  and  then  the  reading  of  this 
point  by  the  scale  of  the  nearly  vertical  lines  is  the 
value  of  \  ks sin  2  6.  Fig.  58  was  constructed  very  care- 
fully, and  can  be  used  with  confidence  ;  and  ordinarily 
will  give  results  as  accurate  as  the  observations. 

Concerning  the  term  (c-\-f)sin  6  of  equation  (13), 
page  194,  notice  that  it  is  always  small,  and  varies  but 
slightly  for  a  considerable  change  in  6  ;  and  therefore 
the  supplemental  table  on  Fig.  58  gives  this  term  with 
sufficient  precision.  The  first  two  values  of  (c  -f  /)  in 
this  table  are  to  be  used  when  the  metre  is  the  unit, 
and  the  last  two  when  the  foot  is  the  unit.  For  values 
of  («•+/)  which  differ  from  those  given,  the  correction 
may  easily  be  interpolated  with  ample  accuracy.  If 
the  particular  value  of  (c  -\-  f)  is  known,  the  correction 
may  be  written  outside  of  the  diagram  between  the 
radii  for  the  different  degrees. 

228.  Another  Geometrical  Diagram.  If  cross-ruled 
paper  is  at  hand,  a  diagram  for  reducing  field  notes 
may  easily  be  constructed  as  follows  :  Establish  a  zero 
at  one  corner  of  the  sheet  and  lay  off  100,  200,  etc., 
along  one  side — say,  the  bottom  of  the  paper, — to  repre- 
sent the  several  values  of  ks,  and  lay  off  10,  20,  etc., 
parallel  to  the  other  side,  to  represent  the  values  sought. 
Obviously  the  scale  of  the  latter  should  be  much  larger 
—say,  ten  or  twenty  times — than  that  of  the  former. 

If  it  is  desired  to  construct  a  table  for  finding  the 
vertical  co-ordinate,  compute,  either  by  the  tables  on 
pages  196-98  or  by  ordinary  trigonometric  tables,  the 
values  of  1,000  -J  sin  2  6  for  the  various  values  of  0, 
and  lay  off  the  distances  vertically  above  the  1,000  point 
on  the  scale  representing  the  values  of  ks]  and  then 
connect  the  points  so  determined  with  the  zero  point. 
The  completed  diagram  will  appear  something  like 
Fig.  59,  page  202.  Since  large  values  of  &sand  #  seldom 


202 


TELEMETERS. 


[CHAP.  X 


occur  together,  the  diagram  may  be  truncated  as  shown, 
which  greatly  extends  the  range  of  a  single  diagram. 
With  cross-ruled  paper  it  is  very  easy  to  construct  a 
diagram  like  that  outlined  in  Fig.  59. 


1000 


m 

FIG.  59. — STADIA  REDUCTION  DIAGRAM. 

229.  SOURCES  OF  ERROR.*     Stadia  work  is  subject  to 
many    of    the    errors    of    ordinary    transit    work    (see 
§§  140-45).     The  additional  errors  peculiar  to  the  stadia 
are  (i)  inclination  of  the  rod,  (2)  error    in  the  value  of 
ks,  and  (3)  error  of  observation. 

230.  Inclination  of  Rod.     To  investigate  the  effect  of 
a   slight   inclination   of   the   rod,  let  ks  =  the  intercept 
when  the  rod  is  vertical,  and  k  s'  =.  the   intercept  when 
the  rod  makes  an  angle  a  with  the  vertical.     Then  k  s  = 
k  s'  cos  a  very  nearly  (see  §  219),  and  the  error  =  k  s' — 
k  s  =  k  s'  (i  —  cos  a)  =  ks'  sin1  %  a  =.  0.000076  k  s'  a'2,  a' 


*  For  general  discussion  of  Cumulative  vs.  Compensating  Errors,  see  §  18. 


V 

ART.   l]  THE    STADIA.  203 

being  in  degrees.  It  is  convenient  to  remember  that  if 
the  rod  inclines  12°  the  resulting  error  in  the  distance 
is  about  one  per  cent.  Notice,  however,  that  ths  error 
increases  as  the  square  of  the  inclination,  which  shows 
the  necessity  of  some  means  of  plumbing  the  rod.  With- 
out some  device  for  keeping  the  rod  vertical  (§  208), 
this  would  undoubtedly  be  the  principal  source  of  error. 
On  the  U.  S.  Lake  Survey,  an  inclined  support  was  used 
to  steady  the  rod  and  prevent  its  leaning  toward  or 
from  the  instrument.  A  light  tripod  is  sometimes  used 
for  the  same  purpose. 

231.  Value  of  ks.    If  the  value  of  ks,  or  the  correspond- 
ing unit  on  the  rod,  is  riot  correctly  determined  the  re- 
sults will  be  incorrect,  although  the  relative  position  of 
points  will  be  right.     The  correct  value  of  ks  is  easily 
found  (see  §§  214-16),  and  with  the  methods  of  attach- 
ing the  hairs  described  in  §  205  there  is  but  little  danger 
of  its  changing. 

232.  Errors  of  Observation.     The  principal  item  under 
this  head  is  the  inaccuracy  in  estimating  the  position  of 
the  hair  on  the  rod.     The  amount  of  this  error  depends 
upon  the  size  of  the  space  on  the  rod  corresponding  to 
a  unit  on  the  ground,  and  upon  the  form  of  graduation. 
For  those  rods  with  which  only  one  side  of  the  hair  is 
observed,  this  error  is  still  further  increased  by  the  un- 
certainty due  to  the  thickness  of  the  hairs,  arid  to  any 
inequality  in  their  thickness  (§  268).     This  element  of 
uncertainty  does  not  exist  with  the  graduations  shown 
in  Figs.  49-52  (page  180). 

Imperfect  focusing,  either  of  rod  or  hairs,  is  a  source 
of  error,  because  it  is  only  when  both  are  in  focus  at  the 
same  time  that  the  assumed  relations  exist.  If  the  rod 
is  not  in  focus,  the  image  covers  too  much  space,  which 
makes  the  distance  too  small.  If  the  cross  hairs  are  not 
in  focus,  the  distance  will  be  read  too  small  or  too 
great  according  as  the  hairs  are  on  one  side  or  the  other 


264  TELEMETERS.  [CHAP.  X 

of  the  focus.  If  there  is  no  parallax  in  the  telescope 
(§  94),  there  will  be  no  error  from  this  cause. 

The  indistinctness  of  the  image  due  to  an  unsteadi- 
ness of  the  atmosphere  produces  an  error  by  making 
the  image  too  large  and,  therefore,  the  distance  too 
small.  The  only  remedy  is  to  wait  for  better  atmos- 
pheric conditions. 

Another  somewhat  common  error  is  miscounting  the 
reading  so  as  to  make  errors  of  10,  100,  etc.,  feet.  These 
errors  may  be  prevented  by  care,  or  checked  by  double 
readings  ;  or,  instead  of  making  an  entirely  new  read- 
ing, the  two  halves  of  the  visual  angle  may  be  read  and 
their  sum  used  to  check  the  reading  of  the  two  outside 
hairs. 

233.  LIMITS  OF  PBECISION.  The  stadia  is  designed  to 
secure  rapidity  rather  than  accuracy  ;  but  nevertheless 
with  reasonable  care  a  considerable  degree  of  accuracy 
may  be  obtained.  The  claim  is  sometimes  made  that 
the  stadia  is  more  accurate  than  the  chain;  but  from  the 
nature  of  the  principles  involved,  it  can  not  be  under 
equally  favorable  conditions,  although  under  some  cir- 
cumstances the  stadia  is  more  accurate  than  the  chain. 
The  degree  of  precision  is  dependent  upon  the  mag- 
nifying power  of  the  telescope,  the  length  of  sight,  and 
the  ratio  of  the  space  on  the  rod  to  the  corresponding 
space  on  the  ground. 

To  ascertain  the  effect  of  the  magnifying  pow  r,  fight 
of  the  author's  students  determined  a  series  of  k/iown 
distances  with  a  telescope  magnifying  fifteen  times  and 
also  with  a  telescope  magnifying  twenty-five  times,  all 
under  essentially  the  same  conditions.  The  average 
error  in  the  first  case  was  i  in  282,  and  in  the  second  i 

in  333- 

To  ascertain  the  relation  between  the  length  of  sight 
and  the  error,  sixteen  of  the  author's  students  deter- 
mined three  series  of  distances  each,  the  first  being  less 


ART.   l]  tHE    STADIA. 


than  100  feet,  the  second  between  100  and  200  feet,  and 
the  third  between  200  and  300  feet.  The  conditions 
were  essentially  the  same  in  all  the  observations.  The 
average  error  in  the  first  series  was  i  in  182,  in  the 
second  i  in  263,  and  in  the  third  i  in  370.  The  relation- 
ship between  the  error  and  the  distance  depends  upon 
the  graduation,  the  magnifying  power,  and  the  atmos- 
pheric conditions  ;  but  for  each  particular  case  there  is 
probably  a  distance  at  which  the  ratio  of  the  error  to 
the  distance  is  a  minimum.  In  a  trial  made  by  the 
author  for  the  purpose  of  this  recor'd,  with  a  graduation 
like  Fig.  49,  page  180  (i  foot  on  the  rod  corresponding 
to  100  feet  on  the  ground)  and  a  transit  magnifying 
twenty-five  times,  this  ratio  was  a  minimum  at  about 
700  feet. 

234.  The  following  data  show  the  degree  of  accu- 
racy attained  in  actual  practice.  They  are  not  selected, 
but  are  all  that  could  be  discovered  by  personal  in- 
quiry and  by  search  through  engineering  literature. 
Apparently  the  results  were  obtained  with  an  ordinary 
engineer's  transit.  In  comparing  results  we  must  dis- 
tinguish between  the  error  of  simply  finding  the  dis- 
tance by  the  stadia,  and  the  final  error  of  a  series  of 
courses  the  lengths  of  which  were  determined  by  the 
stadia.  The  latter  involves  the  error  of  measuring  the 
angles  as  well  as  that  of  measuring  the  distances. 

To  show  the  degree  of  precision  obtained  in  measur- 
ing horizontal  distance,  we  have  the  following  :  Thre£ 
measurements,  each  of  six  distances,  from  50  to  500  ft., 
made  to  test  the  accuracy  of  the  stadia  measurements, 
the  readings  being  made  with  targets,  show  an  average 
error  for  a  single  sight  of  i  in  1,100.*  On  the  U.  S. 
Lake  Survey,  three  measurements  of  a  base  line  gave 
errors  of  i  in  1,000,  i  in  1,635,  an<^  T  'in  i»888.f  On  the 

*  Van  Nostrand's  Engineering  Magazine,  Vol.  30,  pp.  319  and  476. 
f  Journal  of  the  Franklin  Institute,  Vol  .  49,  p.  74. 


2o6  TELEMETERS.  [CHAP.  X 

Mississippi  River  Survey  the  maximum  discrepancy 
permissible  is  i  in  500,  the  maximum  length  of  sight 
being  1,600  feet. 

Only  the  following  data  can  be  found  concerning  the 
accuracy  with  which  vertical  distances  can  be  deter- 
mined. "Courses  have  been  run  with  no  more  than 
ordinary  care  i  to  6  miles,  over  heights  of  150  to  200 
feet,  in  which  the  final  error  in  height  ranged  from  o  to 
1.5  feet."*  In  the  topographical  survey  of  St.  Louis, 
Mo.,  "the  average  error  of  elevations  was  less  than  0.2 
of  a  foot  per  mile.*'  f  The  "  Instructions  for  Topo- 
graphical and  Hydrographical  Field  Work  on  the  Mis- 
sissippi River,"  issued  by  the  Mississippi  River  Com- 
mission, limit  the  maximum  permissible  error  to  i  foot, 
the  maximum  length  of  sight  being  i, 600  feet. 

Concerning  the  final  error  of  a  series  of  courses,  the 
lengths  of  which  were  determined  by  the  stadia,  we 
have  the  following :  "  The  stadia  was  used  for  getting 
the  topography  of  some  densely  wooded  timber  land 
in  the  summer  of  1863.  The  courses  were  so  run  as  to 
connect  points  of  triangulation  from  i  to  4  miles  apart. 
The  distances  from  point  to  point  along  the  courses 
ranged  from  100  to  500  feet.  The  latitudes  and  de- 
partures were  subsequently  computed  with  the  object 
of  finding  the  error  of  stadia  measurements.  The  re- 
sults obtained  were  about  i  in  800,  i  in  1,000,  and  i  in 
1,100."  I  On  the  U.  S.  Lake  Survey,  in  computing  the 
co-ordinates  of  stadia  work  for  1875,  the  length  of  one 
hundred  and  forty-one  lines  varying  between  3,200  feet 
and  22,000  feet  (mean  8,080  feet)  were  compared  with 
the  lengths  determined  by  triangulation  or  chaining, 
and  the  average  error  was  found  to  be  i  in  649.  The 


*  Journal  of  the  Franklin  Institute,  Vol.  49,  p.  74. 

t  The  Technograph,  No.  5  (1890-91),  p.  13. 

}  Journal  of  the  Franklin  Institute,  Vol.  49,  p.  74. 


ART.  l]  THE    STADIA.  207 

maximum  error  permissible  was  i  in  300.*  In  a  survey 
of  the  Red  River  of  the  South,  conducted  by  the  U.  S. 
Army  Engineers,  extending  over  about  70  miles,  the 
stadia  work  and  chaining  were  checked  at  eight  points, 
and  showed  an  average  difference  of  i  in  430.  f  In  the 
topographical  survey  of  St.  Louis,  Mo.,  "the  error  of 
closure,  after  making  corrections  for  inclination  and 
graduation  of  the  rods,  was  about  i  in  800."  J  Without 
these  corrections  the  error  was  about  i  in  500.  Eigh- 
teen observations  were  made  on  two  targets  at  dis- 
tances from  50  to  500  feet,  with  a  mean  error  for  the 
mean  of  three  observations  (the  individual  observa- 
tions are  not  recorded)  of  i  in  1,500.^ 

235.  PRACTICAL  HINTS.  To  obviate  the  difficulty  of 
estimating  a  fraction  of  a  division  of  the  rod  for  each 
stadia  hair,  set  one  of  them  at  the  beginning  of  a 
division.  This  will  produce  a  slight,  but  generally  in- 
appreciable, error  in  the  vertical  co-ordinate.  With  long 
sights  or  small  angles,  it  is  still  more  convenient  and 
sufficiently  accurate,  to  set  one  of  the  hairs  upon  one 
of  the  more  prominent  divisions — as,  for  example,  the 
even  foot-mark.  By  remembering  that  either  hair  may 
be  moved  up  or  down,  and  by  moving  the  telescope  so 
as  to  produce  the  least  displacement,  this  method  of 
placing  one  of  the  hairs  at  an  even  division  can  fre- 
quently be  used  without  any  error,  and  will  greatly 
facilitate  the  work. 

If  not  enough  of  the  rod  is  visible  to  read  both  side 
hairs,  read  first  with  one  side  hair  and  the  middle  one, 
and  then  with  the  middle  one  and  the  other  side  hair; 
and  then  add  the  two  intercepts  and  compute  the  hori- 

*  Professional   Papers,  Corps  of  Engineers,  U.  S.  A.,  No.  24— Primary 
Triangulation  U.  S.  Lake  Survey — p.  34. 

t  Report  of  Chief  of  Engineers,  U.  S.  A.,  1873,  p.  638. 

%  The  Technograph,  No.  5  (1890-91),  p.  12. 

Tf  Van  Nostrand's  Engineering  Magazine,  Vol.  30,  p.  476. 


208  TELEMETERS.  [cHAP.  X 

zontal  and  vertical  co-ordinates  with  the  mean  of  the 
corresponding  vertical  angles. 

In  traversing  with  the  stadia  (§  137  or  §  184),  check  the 
work  by  reading  the  rod  and  the  vertical  angle  on  both 
fore-sights  and  back-sights.  Frequently  the  work  can 
also  be  checked  by  locating  from  each  instrument  sta- 
tion a  point  nearly  in  line  between  them  ;  when  the 
sum  of  the  two  distances  to  the  object  should  be  equal 
to  the  observed  distance  between  the  two  instrument 
stations.  Notice  that  this  check  is  entirely  independent 
of  the  check  by  back-sights  and  fore-sights. 

If  the  target  is  not  visible,  owing  to  brush,  etc.,  sight 
at  any  portion  of  the  rod  that  is  visible  and  note  the 
point  covered  by  the  central  visual  ray.  Make  a  record 
of  this  fact,  and  the  next  time  the  rod  is  visible  through- 
out its  entire  length  determine  the  correction  for  the 
former  reading. 

In  observing  for  azimuth  have  the  edge  of  the  rod 
turned  toward  the  instrument. 

2,36.  The  many  advantages  of  stadia  measurements 
in  surveying  need  not  be  dwelt  upon,  as  they  are  self- 
evident  .to  those  acquainted  with  the  principles.  In 
broken  country,  the  stadia  can  be  readily  used  where 
the  use  of  the  chain  is  not  practicable.  The  stadia  is 
specially  applicable  to  topographical  surveying,  to  the 
topographical  work  of  a  railroad  survey,  and  to  deter- 
mining the  lengths  of  sights  in  leveling. 

The  stadia  may  also  be  used  for  underground  sur- 
veying, where  the  chaining  is  peculiarly  disagreeable 
and  difficult.  For  this  kind  of  work  there  is  substituted 
for  the  rod  a  shallow  box,  lighted  on  the  inside,  with  a 
glass  front  on  which  the  graduation  is  painted.* 


*  Journal  of  the  Franklin  Institute,  Vol.  55,  pp.  384-87. 


ART.   2]  THE    GRADIENTER. 


ART.  2.    THE  GRADIENTER. 

237.  The   gradienter   is    a   tangent    screw   having   a 
micrometer  head,  attached  to  the  horizontal  axis  of  the 
telescope  for  the  purpose  of  measuring  a  vertical  angle 
in  terms  of  its  tangent.     For  a  figure  and  description  of 
the  gradienter,  see  Fig.  28,  page  101. 

238.  THE  GRADIENTER  AS  A  LEVELING  INSTRUMENT. 
As  its  name  indicates,  the  most  important  use  of  the 
gradienter  is  in  locating  grades  in  surveying  railroads, 
irrigating  ditches,  etc.     To  locate  a  grade  of,  for  ex- 
ample, 1.85  per  cent,  i.e.,  1.85  feet  per  100  feet,  bring  the 
telescope  level,   and   read   the   head   of   the  gradienter 
screw;  and  then,  if  the  screw  is  graduated  so  that  one 
revolution  corresponds  to  i  foot  at  100  feet,  turn  the 
screw  1.85  revolutions.     The  line  of  sight  will  then  have 
a  grade  of  1.85  per  cent  up  or  down  according  to  the 
way  the  screw  was  turned;  and  by  setting  a  target  on 
a  rod  at  the  height  of  the  horizontal  axis  of  the  tele- 
scope, the  ground  corresponding  to  this  grade  may  be 
easily  found.     By  an  obvious  modification  of  the  above 
process  this  device  may  be  employed  to  determine  the 
grade  of  any  particular  slope. 

The  gradienter  is  also  very  convenient  in  leveling  up 
and  down  hills,  over  logs,  etc.,  when  extreme  accuracy 
is  not  required. 

Notice  that  to  use  the  gradienter  as  a  leveling  instru- 
ment requires  the  direct  measurement  of  the  horizontal 
distances  with  a  chain  or  tape. 

239.  THE  GRADIENTER  AS  A  TELEMETER.    The  gradi- 
enter may  also  be  used  to  measure  distances,  in  either 
of  two  ways:    (i)  By  measuring  the  space  on   the   rod 
passed  over  by  the  line  of  sight  for  a  given  number  of 
revolutions  of  the  screw;  or  (2)  by  observing  the  num- 


210  TELEMETERS.  [CHAP.  X 

ber  of  revolutions  required  to  carry  the  line  of  sight 
over  a  constant  space  on  the  rod.  The  first  is  the  more 
rapid,  particularly  for  short  sights;  and  the  second  is 
the  more  accurate,  particularly  for  long  sights,  since  the 
observations  may  be  made  on  targets.  Notice  that  there 
is  not  the  same  objection  to  the  use  of  targets  with  a 
fixed  intercept  as  with  a  variable  intercept  (§  207). 

240.  Constant  Number  of  Revolutions  and  Variable 
Intercept.  Line  of  Sight  Perpendicular  to  Rod.  Let  D  = 
the  distance  from  one  extremity  of  the  intercept  to  the 
horizontal  axis  of  the  telescope;  J7=  the  horizontal 
distance  to  be  determined  ;  V '=  the  vertical  distance  to 
be  determined;  ^=the  length  of  the  measured  hori- 
zontal base;  S=  the  space  on  the  vertical  rod  passed 
over  by  one  revolution  of  the  gradienter  screw,  at  the 
distance  B  ;  s  =  the  space  for  one  revolution,  at  the 
distance  D.  Then  by  the  principle  of  similar  triangles 

B  :  D::S:  j;  or 

-0  =  1* •     (i4) 

Usually  gradienters  are  so  made  that  a  single  revolu- 
tion of  the  screw  carries  the  hair  over  i  foot  at  a  distance 

T> 

of  ioo  feet;  that  is,  ordinarily  --  in.equation  (14)  is  equal 

o 

to  ioo.  With  some  forms  of  instruments  two  revolutions 
are  required  to  carry  the  line  of  sight  over  i  foot  at  a 
distance  of  ioo  feet.  In  either  case  the  gradienter  for- 
mula becomes 

Z>ft.  =  loojft (15) 

This  is  the  fundamental  equation  for  the  gradienter, 
and  corresponds  closely  with  the  fundamental  equation 
for  the  stadia — see  equation  (2),  page  184. 


ART.   2] 


THE    GRADIENTER. 


211 


241.  Inclined  Line  of  Sight  and  Vertical  Rod.  It  has 
already  been  shown  (§  218)  that  it  is  better  to  keep  the 
rod  vertical  even  though  the  line  of  sight  is  inclined. 
In  Fig.  60,  AE  represents  the  intercept  on  the  vertical 


FIG.  60. 


rod ;  BE  the  intercept  perpendicular  to  the  lower  visual 
ray;  6  —  the  angle  of  elevation  of  the  lower  visual  ray; 
and  a  =  the  visual  angle. 

From  the  triangle  AEB  we  get 


BE  _  cos  (6  -f-  a)  _  cos  6  cos  a  —  sin  6  sin  a 
AE  ~         cos  a  cos  a 


(16) 


BE  —AE  (cos  6  -  sin  Q  tan  a).   \.     .     (17) 
BE  corresponds  to  s  of  equation  (15),  and  hence 
D  ft.  =  100  BE  =  100  AE  (cos  0  —  sin  0  tan  or).  .     (18) 

Since  ^fi"  =  J  and  tan  «  =  —  =  —  —  (§  240),  the  preceding 

Jj         IOO 

equation  may  be  written  thus: 

D  i t.  =  s  ft.  (100  cos  0  —  sin  0).       .     . 


212  TELEMETERS.  [CHAP.   X 

Notice  that  equation  (19)  was  deduced  for  the  case  in 
which  6  represented  an  angle  of  elevation.  If  6  repre- 
sents an  angle  of  depression,  equation  (19)  is  still  correct 
provided  the  angle  is  measured  to  the  upper  extremity 
of  the  intercept.  But,  for  obvious  reasons,  it  is  better 
to  measure  0  always  to  the  same  extremity  of  the  in- 
tercept —  say,  the  lower,—  in  which  case  equation  (19) 
may  be  used,  if  for  angles  of  depression  6  be  considered 
as  the  observed  angle  minus  the  visual  angle,  i.e.,  if  6  is 
an  angle  of  depression,  subtract  34'  from  it  before  in- 
serting it  in  equation  (19). 

242.  From  equation  (19)  we  easily  get 

&it.=s  ft.  (100  cos2  8  -  i  sin  2  0)*     .     (20) 
H  ft.  =  100  s  ft.  —  100  s  ft.  sin8  6  —  %  s  f  t.  sin  2  6*     (21) 

Equations  (20)  and  (21)  correspond  to  equations  (10) 
and  (n),  page  192,  and  may  be  reduced  in  the  same 
way  —  see  §  224  and  §  226.  Frequently  the  last  term 
in  each  may  be  neglected. 

243.  To  determine  the  vertical   co-ordinate  with   the 
gradienter,  place  a  target  on  the  rod  at  a  distance  from 
its  foot  equal  to  the  height  of  the  horizontal  axis  of  the 
instrument  above  the  reference  point,  and  make  the  tar- 
get the  lower  extremity  of  the  intercept. 

From  equation  (19)  we  easily  get 


.  =  s  ft.  (100  cos  0  sin  6  —sin3  0).*  .     (22) 
Fft.  =  100  s  ft.  -J  sin  2  6  —  s  ft.  sin2  0*  .     (23) 

Equation  (23)  corresponds  to  equation  (13),  page  194, 
and  may  be  reduced  in  the  same  way  —  see  §  224  and 
§  227.  Frequently  the  last  term  can  be  neglected. 

*  If  0  represents  an  angle  of  depression,  subtract  34'  from  it  before  vising  it 
in  th}s  equation. 


ART.  2]  THE    GRADIENTER.  2  13 

244.  Constant  Intercept  and  Variable  Number  of  Revo- 
lutions. Let  D  —  the  perpendicular  distance  from  the 
center  of  the  intercept  to  the  horizontal  axis  of  the  tele- 
scope ;  H  '=  the  horizontal  distance  from  the  rod  to  the 
center  of  the  instrument;  V—  the  vertical  distance 
from  the  point  under  the  instrument  to  the  point  on 
which  the  rod  is  placed  ;  S  =  the  distance  between  tar- 
gets ;  N=  the  number  of  revolutions  required  to  move 
the  line  of  sight  from  one  extremity  to  the  other  of 
the  constant  intercept,  at  the  distance  B\  n  =  the  num- 
ber of  revolutions  required  to  move  the  line  of  sight 
over  the  constant  intercept,  at  the  distance  D. 

Then  by  the  principle  of  the  similarity  of  triangles, 
B  :  D  ::  n  :  N,  or 


(24) 


Ordinarily  gradienters  are  so  made  that  if  B  =  100  feet 
and  S  —  i  foot,  JV  =  i  ;  and  consequently  for  any  value 
of  S  the  number  of  units  in  S  and  Wwill  be  the  same. 
Under  these  conditions,  equation  (24)  becomes 


This  equation  corresponds  to  equation  (2),  page  184, 
and  equation  (15),  page  210,  and  is  the  fundamental 
equation  for  this  method  of  using  the  gradienter. 
Notice  that  *$*,  the  distance  between  the  targets,  can  be 
changed,  as  desired,  to  suit  the  nature  of  the  work. 

245.  From  Fig.  60,  page  211,  we  may  at  once  write 


214  TELEMETERS.  [CHAP.  X 

in  which  8  is  measured  to  the  lower  of  the  targets,  the 
observed  value  being  used  for  angles  of  elevation, 
and  for  angles  of  depression  N  times  34'  is  subtracted 
from  the  observed  value  before  inserting  it  in  the  for- 
mula (see  §  241).  The  numerator  of  equation  (26) 
may  be  found  by  Table  III,  pages  196-98,  or  by  Fig.  57, 
between  pages  200  and  201. 

246.  If  the  bottom  target  is  placed  at  a  distance 
from  the  foot  of  the  rod  equal  to  the  height  of  the  in- 
strument above  the  reference  point,  then 


in  which  #  is  measured  as  in  §  245.  The  numerator  of 
equation  (27)  can  be  found  by  Table  III,  pages  196-98, 
or  by  Fig.  58,  between  pages  200  and  201. 

247.  Stadia  vs.  Gradienter.      It  is  sometimes  claimed 
that  the  gradienter  is  superior  to  the  stadia,  but  this  is 
at   least  doubtful.     When   the  gradienter  is  employed 
with  a  variable  intercept  (§§  240-43),  the  labor  and  time 
required  for  the  observations  are  essentially  the  same 
as  with  the  stadia;  but  the  formulas  —  equations  (20)  or 
(21),  and  (23)  —  are  more  complicated,  particularly  the 
correction   of    6  for  angles  of  depression.     When   the 
gradienter    is    employed    with    a    constant     intercept 
(§§  244-46),  the  time  required  to  make  an  observation  is 
greater  than  with  the  stadia,  and  the  formulas  —  equa- 
tions (26)  and  (27)  —  are  considerably  more  complicated. 

248.  LIMITS  OF  PRECISION.     The  gradienter  as  a  dis- 
tance-measurer is  not  as  accurate  as  the  stadia.     This 
is  chiefly  due  to  the  fact   that  with   the  former,  two 
observations  must  be  made  upon  each   point  and  the 
instrument   is    liable    to    be    disturbed    between    these 
observations.     Further,  changes  of  atmospheric  refrac- 
tion occur  quickly,  so  that  there  is  more  risk  of  error 


ART.  2]  THE   GRADIENTER. 


from  two  separate  observations  than  if  they  were  made 
simultaneously,  as  with  the  stadia. 

In  using  the  gradienter,  to  eliminate  any  back-lash  or 
lost  motion  in  the  screw  it  is  best  to  finish  the  setting 
with  a  motion  always  in  the  same  direction. 

249.  VEKTICAL  CIRCLE  AS  A  GRADIENTER.      The  ver- 
tical circle  may  be  employed  as  a  gradienter  to  lay  off 
grades,  or  as  a  telemeter  to  determine  the  horizontal 
and  vertical  co-ordinates  of  a  point. 

250.  To  Measure  Grades.     To  measure  grades  with  the 
vertical  circle,  it  is  only  necessary  to  change  the  degrees 
and  minutes  of  arc  into  feet  per  hundred  feet,  by  means 
of  a  table  of  natural  tangents.     For  example,  if  we  de- 
sired  to  run  a  grade  line  of  one   in  a  hundred,  />.,  a  i 
per  cent  grade,  we  look  in  a  table  of  natural  tangents 
and  find  the  angle  whose  tangent  is  o.oi,  and  set  it  off 
on  the  vertical  circle.     Similarly  to  run  a  1.85  per  cent 
grade,  i.e.,  a  grade  of  1.85  ft.  per  100  ft.,  we  would  find 
the  angle  whose  tangent  is  0.0185. 

If  the  angle  of  inclination  of  a  particular  slope 
has  been  measured,  and  we  desire  to  express  it  in  feet 
per  hundred,  we  have  only  to  find  the  natural  tangent 
of  the  observed  angle.  For  example,  if  the  slope  has 
an  inclination  of  43',  an  inspection  of  a  table  of  natural 
tangents  shows  that  the  grade  is  1.25  per  cent. 

251.  To  Measure  Distances.     To  use  the  vertical  circle 
as  a  telemeter,  notice  that  if  the  angle  of  elevation  or 
depression  of    two  points    in  the  same  vertical  be  ob- 
served, the  distance  between  the  points  is  the  difference 
between  the  natural  tangents  of  the  observed  angles. 

To  deduce  a  formula  for  this  case,  let  If  =  the 
horizontal  distance  from  the  instrument  to  the  rod  ; 
V  '=  the  vertical  distance  from  the  point  on  the  ground 
under  the  instrument  to  the  point  upon  which  the  rod 
is  placed  ;  a  =  the  larger  of  the  two  observed  angles  ; 

=  the    smaller  of  the   two  angles  ;  and  s  =  the  dis- 


2l6  TELEMETERS.  [cttAP.  X 

tance  on  the  rod  between  the  two  points  sighted  at. 
Then,  if  a  and  ft  have  the  same  sign,  t.e.,  if  both  are 
angles  of  elevation  or  angles  of  depression, 

H=  -  ~  --  -5.     .     .    .  (28) 

tan  a  —  tan  ft 

If  a  and  ft  do  not  have  the  same  sign, 

H  =  -  -  ---  3.  (29) 

tan  a  -f-  tan  ft 

If  the  target  to  which  ft  is  measured  is  placed  at  a 
distance  from  the  foot  of  the  rod  equal  to  the  height  of 
the  instrument  above  the  reference  point, 

F  =  ^tan/?;     ......     (30) 


tan  a  T  tan  / 

ART.  3.     VARIOUS  TELEMETERS. 

252.  Eckhold's  Telerneter  consists  of  a  telescope  and 
micrometer-microscope  firmly  connected  at  right  angles 
with  each  other,  and  both  turning  together  in  the 
same  vertical  plane.  Below  the  rotation  axis,  in  the 
plane  of  the  telescope,  is  placed  a  finely  graduated 
scale,  which  is  read  through  the  microscope.  The  in- 
tercept on  the  rod  is  of  a  constant  length.  To  use  this 
telemeter  the  telescope  is  directed  to  the  top  of  the  rod 
and  the  microscope  read;  the  telescope  is  next  directed 
to  the  bottom  of  the  rod  and  the  microscope  is  read 
again.  The  horizontal  and  vertical  distances  are  com- 
puted by  formulas  essentially  the  same  as  equations 
(28)  or  (29),  and  (31)  above. 

,  Extravagant  claims  are  made  for  this  instrument,  but 
they  are  not  sustained  by  experience.  It  is  inferior  in 
accuracy  to  the  stadia. 


ART.  3]  VARIOUS    TELEMETERS.  21^ 

253.  Gautier's  Telemeter  consists  of  a  combination  of 
two  mirrors  and  a  prism,   such  that  when  two  succes- 
sive observations  are  made  from  two  points  (a  few  feet 
apart)  upon  a  third,  the  distance  to  the  third  point  is  the 
product  of  a  factor  read  from  the  instrument,  and  the 
distance  between  the  two  points  of  observation. 

"The  accuracy  of  this  instrument  is  extraordinary. 
With  a  base  of  20  meters  [66  feet],  the  error  for  dis- 
tances below  a  kilometer  [3,280  feet]  is  almost  imper- 
ceptible. Distances  from  3  to  6  kilometers  [roughly  2 
to  4  miles],  and  even  more,  have  been  measured  by  it, 
with  bases  of  from  20  to  50  meters  [66  to  164  feet],  with 
a  maximum  error  not  exceeding  one  fourth  of  one  per 
cent."  Obviously  such  accuracy  is  impossible,  i.  Only 
a  low  magnifying  power  is  claimed  for  the  telescope. 
2.  The  observations  are  made  by  the  coincidence  of 
images,  as  in  the  sextant.  3.  The  instrument  is  held  in 
the  hand.  4.  The  error  of  measuring  the  base  is  multi- 
plied in  the  result. 

254.  Adie's,  Smyth's,  Clarke's,  and  Struve's    Telemeters 
each  consist  of  two  mirrors   mounted  on  the  ends  of 
a  rod,  the  direction  of  a  third  point  being  measured 
by  the  coincidence  of  images,  as  with  the  sextant.     In 
Adie's  telemeter  the  base  is  36  inches,  in   Struvejs   75 
inches,  and    the    others    mentioned    have   bases   inter- 
mediate between  these  two. 

It  is  now  generally  conceded  that  the  sextant  is  the 
best  instrument  for  determining  a  distance  when  a 
measured  base  can  be  fixed. 

255.  Other  Telemeters.     There  are  many  other  forms 
of    telemeters,    several    of    which    were    invented    for 
military  use,  but  the  above  illustrate  the  principles  of, 
at  least,  the  most  important  ones.     Only  the  stadia  with 
fixed  hairs  and  the  gradienter  have  any  value  as  engi- 
neering instruments. 


CHAPTER  XI. 
SPIRIT    LEVELS. 

ART.  1.     CONSTRUCTION. 

257.  QUALITIES  DESIRED.  The  main  qualities  to  be 
secured  in  a  spirit-leveling  instrument  are  stability  of 
the  instrument,  defining  and  magnifying  power  of  the 
telescope,  and  delicacy  of  the  level  bubble. 

The  stability  of  the  instrument  depends  mainly  upon 
the  leveling  screws  and  the  manner  of  connecting  the 
instrument  with  the  tripod  head  (see  Chap.  II,  Art. 
2).  The  ordinary  leveling  instrument,  as  well  as  the 
transit,  has  four  foot-screws;  but  as  three  give  more 
stability  and  greater  delicacy,  it  is  more  important 
to  have  three  foot-screws  on  levels  than  on  transits, 
for  it  is  seldom  necessary  that  the  latter  be  exactly 
level^l  128).  Whether  three  or  four  foot-screws  are 
used,  the  distance  between  opposite  screws  should  be 
as  great  as  possible.  The  center,  or  spindle,  should  be 
long  and  hard,  and  well  fitted  in  the  socket.  Obviously 
the  center  of  gravity  of  the  instrument  should  be  as 
near  to  the  tripod  head  as  possible,  although  many 
instruments  are  needlessly  defective  in  this  particular. 

The  optical  qualities  of  the  telescope  have  already 
been  discussed  (see  Chap.  VI). 

However  good  the  other  parts  of  the  instrument,  the 
accuracy  of  the  work  depends  upon  the  sensitiveness  of 
the  bubble.  The  bubble  of  a  leveling  instrument  cor- 

218 


ART.   l  CONSTRUCTION. 


responds  in  importance  with  the  graduation  of  a  transit. 
It  must  be  remembered  that  although  a  sensitive  bub- 
ble may  not  remain  exactly  stationary,  it  will  still  give 
better  results  than  a  sluggish  one  which  shows  no 
movement  when  the  inclination  of  the  instrument  is 
slightly  changed.  The  liquid  employed  in  level  vials 
should  have  a  minimum  adhesion  to  the  glass  so  as  to 
settle  quickly  and  accurately.  Pure  ether  is  best  in  this 
respect,  but  it  has  too  large  a  co-efficient  of  expansion 
to  permit  its  use  in  field  instruments,  which  are  exposed 
to  great  differences  of  temperature.  Pure  alcohol  is 
frequently  employed,  but  a  mixture  of  alcohol  and 
ether  is  generally  considered  best.  The  level  vials 
of  astronomical  instruments  and  of  some  of  the  best 
engineering  field  instruments  are  provided  with  an  air- 
chamber  for  regulating  the  length  of  the  bubble,  in 
which  case  pure  ether  can  be  used.  A  large  air-bubble 
is  more  sensitive  than  a  small  one. 

The  scale  by  which  the  position  of  the  bubble  is  read 
should  be  as  close  to  the  bubble  as  possible,  to  avoid 
parallax  in  reading;  and  therefore  the  graduation  should 
be  upon  the  glass  tube.  The  glass  tube  should  be  pro- 
tected by  a  metal  case,  but  the  former  should  be  so 
fastened  into  the  latter  as  to  be  free  to  expand  and 
contract  with  changes  of  temperature.  The  common 
method  of  fastening  the  vial  in  the  metal  case  with 
plaster  of  Paris  is  inadvisable  for  a  sensitive  bubble, 
since  changes  of  temperature  cause  changes  in  the  cur- 
vature of  the  tube  and  consequently  in  its  sensitive- 
ness. 

258.  CLASSIFICATION.  Spirit-leveling  instruments  may 
be  grouped  in  three  classes.  The  first  includes  all  in- 
struments that  can  be  adjusted  by  reversals.  The  wye 
level,  Fig.  61,  page  221,  is  the  representative  of  this  class. 
The  second  includes  all  leveling  instruments  that  can 
not  be  adjusted  by  reversals.  The  dumpy  level,  Fig. 


226  SPIRIT    LEVELS.  [CHAP.  XI 

63,  page  223,  is  the  representative  of  this  class.     The 
third   includes  all  instruments  whose  errors  of  adjust- 
ment may  be  eliminated  by  double  observations.     Fig. 

64,  page  225,  is  a  representative.     Instruments  of  this 
class  are  commonly  called  levels  of  precision, — sometimes 
geodesic  levels; — and  are  used  when  extreme  accuracy 
is  sought. 

259.  Wye  Level.  The  instrument  shown  in  Fig.  61,  is 
named  the  wye,  or  Y,  level  from  the  form  of  the  supports 
of  the  telescope.  A  section  of  a  wye  level,  by  a  different 
maker  than  that  of  Fig.  61,  is  shown  in  Fig.  62,  page 
222.  The  wye  level  is  used  far  more  than  any  other 
by  American  engineers.  Its  distinguishing  characteris- 
tic is  that  the  telescope  may  be  revolved  about  its  own 
axis,  and  turned  end  for  end  in  its  bearings.  The  only 
advantage  of  this  construction  is  that  it  facilitates  the 
adjustment  of  the  instrument;  while,  on  the  other  hand, 
owing  to  the  nature  of  the  construction  the  instrument 
does  not  hold  its  adjustment  well.  "  The  wye  level  is 
easily  adjusted,  and  nearly  always  needs  it." 

Notice  that  the  instrument  shown  in  Fig.  61  has  an 
inverting  eye-piece,  and  that  in  Fig.  62  an  erecting  eye- 
piece. The  latter  is  much  more  common,  but  the  for- 
mer is  the  better  (see  §  78). 

To  adjust  the  telescope,  it  must  be  free  to  revolve 
in  the  wyes  about  its  own  axis;  but  in  using  the  in- 
strument, it  is  necessary  that  the  vertical  hair  should 
be  vertical.  Therefore,  when  the  telescope  is  fastened 
in  the  wyes  ready  for  use,  there  should  be  some  means 
of  knowing  that  the  vertical  hair  is  truly  vertical. 
For  this  purpose  some  makers  place  a  mark  upon 
the  wye  and  another  upon  the  collar  on  the  telescope; 
while  others  have  a  device  (of  which  there  are  several 
forms)  such  that  the  clip  passing  over  the  telescope  at 
the  top  of  the  wyes  can  not  be  fastened  until  the  tele- 
scope is  in  a  particular  position.  One  of  these  devices 


ART. 


CONSTRUCTION. 


221 


222 


SPIRIT    LEVELS. 


[CHAP.  XI 


ART.    l] 


CONSTRUCTION. 


223 


is  shown  in  Fig.  61,  page  221,  on  the  inner  side  of  each 
clip. 

260.  Dumpy  Level.     This  name  is  given  to  that  form 
of  leveling  instrument  in  which  the  telescope  is  attached 


FIG.  63.— DUMPY  LEVEL. 

to  the  bar  in  such  a  way  as  not  to  admit  of  its  rotation 
around  its  own  axis,  nor  to  allow  of  its  reversion  end  for 
end.  The  telescope  is  usually  inverting,  and  therefore 
shorter  than  the  one  commonly  used  on  leveling  instru- 
ments; hence  the  name  dumpy  level.  A  very  good  form 


224  SPIRIT    LEVELS.  [CHAP.  XI 

of  this  instrument  is  shown  in  Fig.  63,  page  223.  English 
engineers  use  the  dumpy  level  almost  exclusively.  In 
the  English  books  it  is  sometimes  called  the  Troughton 
level,  from  the  first  manufacturer;  and  sometimes  the 
Gavatt  level,  in  honor  of  Gavatt's  improvements. 

In  construction  the  dumpy  is  more  simple  and  com- 
pact than  the  wye  level,  but  is  less  convenient  to  adjust. 
It  retains  its  adjustments  better  than  the  wye  level, 
which  is  an  important  item  in  practice.  If  equally  well 
made,  it  will  do  as  accurate  work  as  the  more  elaborate 
and  more  expensive  wye  level.  The  dumpy  level  usu- 
ally has  the  inverting  telescope,  which  gives  it  an  advan- 
tage over  the  ordinary  wye  level  (see  §  78). 

261.  Levels  of  Precision.     The  distinguishing  charac- 
teristic of  this  class  of  levels  is  that  the  telescope  may 
be  revolved  about  its  own  axis,  and  the  level  may  be 
reversed  end  for  end  independently  of  the  telescope. 
This  construction  enables  the  observer  to  eliminate  all 
errors  of  adjustment,  by  making  a  double  observation. 

There  is  considerable  variety  in  the  form  of  this 
class  of  levels,  but  only  two  have  been  used  to  any 
appreciable  extent  in  this  country:  the  Swiss  or  Kern 
level,  by  the  Lake  Survey  and  Mississippi  River  Com- 
mission; and  a  modification  of  the  Vienna  or  Stampfer 
level,  by  the  Coast  and  Geodetic  Survey.  The  con- 
struction of  the  two  is  similar,  hence  only  the  latter 
will  be  described  here.  The  former  is  described  in 
Professional  Papers,  Corps  of  Engineers,  U.  S.  A.,  No. 
24 — Primary  Triangulation  U.  S.  Lake  Survey, — p.  597, 
and  also  in  Report  of  Chief  of  Engineers,  U.  S.  A.,  for 
1877,  p.  1190. 

262.  The  instrument  shown  in   Fig.  64  is    the   level 
of  precision    employed    on   the   U.  S.   Coast  and  Geo- 
detic   Survey.*     The    telescope    may   be    reversed    end 

*  Report  for  1879,  Appendix  No.  15,  pp.  202-11. 


ART.   l]  CONSTRUCTION.  225 

for  end  and  revolved  about  its  optical  axis,  the  two 
positions  in  which  the  horizontal  thread  is  horizontal 
being  definitely  fixed  by  projecting  pins.  The  level 
may  also  be  reversed  end  for  end  independently  of 


FIG.  64. — U.  S.  COAST  SURVEY  LEVEL  OF  PRECISION. 

the  telescope.  One  end  of  the  telescope  and  level 
may  be  raised  and  lowered  by  the  micrometer  screw. 
Near  the  micrometer  is  a  cam  hook,  by  which  the 
weight  of  the  superstructure  may  be  raised  off  the 
micrometer  during  transportation.  Under  the  telescope 
are  two  false  wyes  on  lever  arms  by  which  the  telescope 
may  be  raised  out  of  the  wyes  for  transportation.  The 


226 


SPIRIT    LEVELS. 


[CHAP,  xi 


whole  instrument  is  secured  to  the  tripod-head  by  a 
brass  plate  which  fits  over  the  feet  of  the  leveling 
screws. 

The  aperture  of  the  telescope  is  43  mm.  (if  inches 
nearly),  focal  length  410  mm.  (16^  inches  nearly),  mag- 
nifying power  37.  The  value  of  one  millimeter  of  the 
level  scale  is  1.5"  (radius  =  450  feet).  The  diaphragm 
is  glass,  and  has  one  vertical  and  two  horizontal  lines 
ruled  upon  it.  The  two  horizontal  lines  are  used  as 
stadia  hairs  to  determine  the  length  of  sight. 

The  instrument  including  the  tripod  weighs  45 
pounds.  A  smaller  size  of  this  instrument,  weight  23 
pounds,  is  also  used. 

263.  The  following  table  contains  some  of  the  im- 
portant information  concerning  the  instruments  used 
on  four  national  surveys.* 

TABLE  IV. 
DATA  CONCERNING  STANDARD  LEVELS  OF  PRECISION. 


Great 
Britain. 

India. 

Switzer- 
land. 

France. 

BUBBLE  TUBE  : 

Radius  of  curvature  . 

360  ft. 

540  ft. 

600  ft. 

360  ft. 

Movement  of  line  of 

sight    correspond- 

ing to  a  movement 

of  the  bubble  of  o.  I 

inch    

5" 

3i" 

3" 

3"  to  7" 

TELESCOPE  : 

Focal  length    .     .     . 

24  ins. 

21  ins. 

16  ins. 

19  ins. 

Diameter  of  objective 

2^  ins. 

2  ins. 

ii  ins. 

ii  ins. 

Magnifying  power   . 

50 

42 

42 

36 

RODS  : 

Length  

10  ft. 

1  6  ft. 

9.8  ft. 

Graduated  to  ... 

I-IOO  ft. 

I-IOO  ft. 

3-100  ft. 

6-100  ft. 

Read  to  

I—  1,000  ft. 

1—1,000  ft. 

3—1,000  ft. 

3-1,000  ft. 

LENGTH  OF  SIGHT  : 

Maximum     distance 

from  instrument  to 

rod 

aqo  ft 

660  ft. 

OTQ    ft 

J.3O  ft 

•    JO**1  *•*•• 

JJU    1L. 

4OU  »*« 

*  Proc.  Inst,  of  C.  E.r  Vol.  44,  p.  181. 


ART  2]  LEVELING    RODS.  227 


ART.  2.     CONSTRUCTION  OF  LEVELING  RODS. 

264.  The  line  of  sight  of  the  level  is  horizontal,  and 
the  distance  of  points  below  this  line  is  measured  by  a 
graduated    rod    held    vertically.     Rods    are    generally 
graduated  to  read  feet  and  decimals  of  a  foot,  and  oc- 
casionally to  feet,  inches,  and  fractions  of  an  inch;    and 
rods  graduated    according   to    the    metric    system   are 
sometimes  used  in  this  country. 

There  are  two  classes  of  leveling  rods:  (i)  target  rods, 
those  having  a  target  which  is  moved  into  the  plane  of 
sight,  its  position  being  read  by  the  rod-man;  and  (2) 
self-reading  or  speaking  rods,  those  having  a  graduation 
such  that  the  position  of  the  intersection  of  the  line  of 
sight  and  the  rod  can  be  read  with  the  telescope.  With 
the  self-reading  rod  the  rod-man  has  only  to  hold  the 
rod  vertical. 

265.  TAEGET  RODS.     New  York  Rod.     The  target  rod 
most   frequently  used  in  this  country  is  the  New  York 
rod,  shown  in  Fig.  65,  page  230.     This  rod  usually  con- 
sists  of  two  pieces  of  maple  or  satinwood   sliding  one 
upon  the  other,  the  same  end  always  being  held  on  the 
ground,  and  the   graduations   starting  from    this   end. 
The  graduations  are  to  tenths  and  hundredths  of  a  foot, 
the   tenth    figures    being    marked   with   a  black  figure 
and  the  feet  with  a  larger  red  figure.     The  target  carries 
a  vernier  reading  to  thousandths  of  a  foot  (see  Fig.  10, 
page  67). 

The  front  surface,  i.e.,  the  one  on  which  the  target 
moves,  reads  to  6.5  feet.  When  a  greater  height  is 
required,  the  horizontal  line  of  the  target  is  fixed  at  6.5 
feet,  and  the  upper  half  of  the  rod,  carrying  the  target, 
is  slid  upward,  and  the  reading  is  obtained  by  a  vernier 
on  the  side  of  the  rod.  The  rod,  when  extended,  may 


228  SPIRIT    LEVELS.  [CHAP.  XI 

be  fastened  in  that  position  by  means  of  a  clamp  at  the 
lower  end  of  the  upper  piece. 

This  rod  is  also  made  in  three  and  sometimes  four 
pieces.  These  forms  are  recent  modifications  to  make 
the  rod  shorter  when  closed  and  longer  when  extended. 
The  three-piece  rod  is  5  feet  long  when  closed  and  14 
feet  long  when  extended.  The  four-piece  rod  when 
closed  is  5  feet  long,  and  when  extended  is  16  feet 
long. 

266.  Boston  Rod.     The  Boston  Rod,  Fig.  66,  page  229, 
is  formed  of  two  pieces  of  mahogany  or  baywood,  each 
about   6   feet   long  and   sliding   easily  by  each  other  in 
either  direction.     One  piece  is   furnished   at   each   end 
with   a  clamp   and   also  a  vernier  ;   the  other,  or  front 
piece,  carries  the  target,  and  has  on  each  edge  a  strip 
of    satinwood    inlaid,    upon    which    divisions    of    feet, 
tenths,   and  hundredths  are  marked.     The  target  is  a 
rectangle  of  wood  fastened  on  the  front  half. 

When  a  reading  of  less  than  6  feet  is  desired,  the  rod 
is  placed  target-end  down,  and  the  piece  carrying  the 
target  is  raised.  When  a  reading  of  more  than  6  feet  is 
desired,  the  rod  is  placed  target-end  up,  and  the  piece 
carrying  the  target  is  raised,  the  reading  being  taken 
from  the  other  vernier.  This  rod  is  very  convenient 
owing  to  its  extreme  lightness,  but  the  parts  are  too 
frail  to  endure  rough  usage,  and  therefore  engineers 
have  generally  given  the  preference  to  heavier  and  more 
substantial  rods. 

267.  Philadelphia  Rod.     The   Philadelphia  Rod,  Fig. 
67,  page  229,  is  a  self-reading  rod  which  is  fitted  with  a 
target.     The  rod  is  made  of  two  strips  of  cherry,  each 
about  |  inch   thick  by  ij   inches  wide   and  7  feet  long, 
connected  by  two  metal  sleeves,  the  lower  one  of  which 
has   a   clamping  screw  for  fastening   the  two  parts  to- 
gether when    the   rod   is   raised   for  a  greater   reading 
than  7  feet.     Both  sides  of  the  back  strip  and  one  side 


ART.   2] 


LEVELING    RODS. 


229 


FIG.  65. 
NEW  YORK  ROD. 


FIG.  66. 
BOSTON  ROD. 


FIG.  67. 
PHILADELPHIA  ROD. 


230  SPIRIT    LEVELS.  [CHAP.  XI 

of  the  front  one  are  recessed  one  sixteenth  of  an  inch 
below  the  edges.  These  depressed  surfaces  are  painted 
white,  and  divided  into  feet,  tenths,  and  hundredthsof  a 
foot,  according  to  the  general  principle  illustrated  in 
Fig.  72,  page  233.  The  front  piece  is  graduated  from 
the  bottom  upward  to  7  feet,  the  front  surface  of  the 
rear  half  from  7  to  13  feet,  also  from  the  bottom  up- 
ward ;  and  the  back  surface  of  the  rear  half  is  figured 
from  7  to  13  feet,  from  the  top  downward. 

This  rod  may  be  used  either  as  a  target  rod  or  as  a 
self-reading  rod.  The  target,  usually  painted  as  Fig, 
69,  page  231,  carries  a  scale  (not  a  vernier)  one  tenth 
of  a  foot  long  divided  to  hundredths  and  half-hun- 
dredths  of  a  foot,  by  which  the  rod  is  read  to  half- 
hundredths  of  a  foot.  When  used  as  a  target  rod, 
for  readings  of  less  than  7  feet  the  target  is  moved  up 
and  down  the  front  piece  ;  and  for  readings  greater 
than  7  feet  the  target  is  set  at  7  feet  and  the  back 
piece  is  run  up,  the  reading  then  being  obtained  by 
a  scale  upon  the  back  piece.  When  used  as  a  self- 
reading  rod,  the  observer  notes,  through  the  telescope, 
the  point  on  the  rod  covered  by  the  cross  hair.  For 
readings  greater  than  7  feet,  the  rear  piece  is  fully  ex- 
tended, the  whole  front  of  the  rod  then  becoming  a  self- 
reading  rod  13  feet  long. 

268.  The  Target.  This  is  a  piece  of  brass  or  iron 
which  can  be  moved  up  or  down  the  rod,  or  clamped  in 
any  position.  It  carries  a  scale  or  vernier  to  subdivide 
the  least  space  on  the  rod.  The  face  of  the  target 
should  be  painted  of  such  a  pattern  that  it  may  be  .pre- 
cisely bisected  by  the  horizontal  cross  hair.  Some  of 
the  many  varieties  are  shown  in  Figs.  68,  69,  70,  and  71, 
page  231. 

Fig.  68  depends  upon  the  nicety  with  which  the  eye 
can  determine  whether  a  line  bisects  an  angle,  which 
can  be  done  very  accurately  by  noticing  the  relative  po- 


ART.   2] 


LEVELING    RODS. 


sition  of  the  two  points  formed  on  opposite  sides  of  the 
hair.  For  data  on  the  relative  accuracy  of  the  targets 
shown  in  Figs.  68  and  69,866  Tables  I  and  II,  Appendix 
III. 

Fig.    69   is    the    target  of    the   New  York    rod,   and 
essentially  that  of  the  Philadelphia  rod.     The  design  is 


FIG.  68. 


FIG.  69. 


FIG.  70. 


FIG.  71. 


not  good,  because  the  cross  hair  may  be  above  or  be- 
low the  middle  of  the  target  by  its  full  thickness,  as 
magnified  by  the  eye-piece,  without  the  error's  being 
perceptible. 

Fig.  70  is  the  same  in  principle  as  Fig.  68.  For  long 
sights  Fig.  68  is  the  better,  but  for  very  short  sights 
the  diamond  is  larger  than  the  field  of  view.  Fig.  70 


232  SPIRIT    LEVELS.  [CHAP.   Xi 

was  designed  to  obviate  this  defect,  and  is  an  excellent 
target  except  for  very  long  sights. 

Fig.  71  depends  upon  the  accuracy  with  which  the 
eye  can  bisect  a  space.  The  only  objection  to  this  form 
of  target  is  that  for  the  greatest  accuracy  the  width  of 
the  white  band  should  be  proportional  to  the  length  of 
the  sight  ;  and  therefore  if  the  width  is  right  for  short 
sights  it  will  be  too  narrow  for  long  ones.  If  the  length 
of  a  sight  were  constant  this  would  probably  be  the 
best  form  of  target. 

269.  Level  targets  are  usually  painted  red  and  white. 
Black  and   white  are  best   for  visibility  ;  but  red   and 
white    are    most    easily    distinguished     among    trees, 
shadows,  etc.,  and  red  gives  the  stronger  contrast  with 
the  cross  hairs.     Probably  red  and   white  are  the  best 
for  Fig.  69,  and  black  and  white  for  the  other  figures  on 
page  231. 

270.  SELF-HEADING  RODS.     A  self-reading  rod  is  one 
so  graduated  as  to  enable  the  observer  to  note  at  once 
the  reading  of  the  point  which  lies  in  the  line  of  sight. 
The   rod-man   has  only  to  hold    the   rod  vertical  ;   the 
observer  notes   and   records   the  reading.      The    most 
common  rod  of  this  class  is  the  Philadelphia  rod,  which 
may  be  used  as  a  target  rod  also  (§  265).     A  few  of  the 
many  patterns  which  have  been  proposed  for  self-read- 
ing leveling  rods  are  shown  in  Figs.  72-75,  page  233. 

Fig.  72  illustrates  the  principle  of  the  Philadelphia 
rod  (§  267).  The  figures  indicating  feet  are  red,  the 
tenths  black.  The  figures  are  six  hundredths  in  height, 
placed  with  centers  over  the  marks.  Sometimes  the 
bottom  of  the  figure  is  placed  on  the  line,  in  which  case 
the  figures  are  eight  hundredths  high.  The  angles  of 
the  figures  indicate  fractions  of  tenths.  The  gradua- 
tion given  is  suitable  for  short  sights;  and  for  longer 
ones  the  figures  are  made  larger,  and  only  those  for  the 
even  tenths  are  marked.  The  advantages  claimed  for 


ART.  2]  CONSTRUCTION    OF    LEVELING    RODS. 


233 


FIG.  72. 


FIG.  73. 


FIG.  74. 


FIG.  75. 


234  SPIRIT    LEVELS.  [CHAP.   XI 

this  general  form  of  graduation  are  distinctness  and 
visibility  ;  but  in  neither  of  these  respects  is  it  as  good 
as  Figs.  74  or  75,  or  as  the  stadia  rods  shown  in  Figs. 
49-52,  page  1 80. 

Fig.  73  shows  the  graduation  of  the  Francis  rod,* 
which  is  a  complete  graduation  to  hundredths  without 
visual  division,  and  can  be  read  without  counting.  The 
foot-marks  are  red,  and  larger  than  the  tenths.  It  is 
best  adapted  to  short  sights. 

Fig.  74,  in  its  essential  features,  is  the  graduation  of 
the  favorite  British  level  rod.  The  graduation  shown  is 
to  tenths,  the  hundredths  to  be  estimated.  The  gradua- 
tion of  the  standard  British  rod  consists  of  a  number  of 
black  lines  on  a  white  ground,  one  hundredth  of  a  foot 
wide  and  one  hundredth  of  a  foot  apart.  The  lines  in- 
dicating the  tenths  are  longer  than  the  others,  and  each 
alternate  tenth  is  numbered.  This  graduation  is  very 
confusing  to  read.  The  one  shown  in  Fig.  74  is  de- 
signed on  the  principle  that  it  is  better  to  estimate  the 
hundredths  than  to  read  them  from  a  finely  divided 
scale.  This  form  of  graduation  reduces  the  difficulty 
of  counting  a  number  of  small  divisions,  and  also  the 
possibility  of  gross  mistakes.  Fig.  74  has  greater  dis- 
tinctness and  visibility  than  the  standard  British  rod, 
but  it  must  be  condemned  for  the  same  reason  that  the 
quadrant  target  of  the  New  York  rod  was  condemned 
in  §  268. 

The  principle  of  the  Texas  rod,f  Fig.  75,  is  of  fre- 
quent application  in  making  self-reading  rods.  It  is 
superior  to  that  of  Fig.  74  in  that  it  obviates  the  error 
due  to  the  thickness  of  the  cross  hair.  The  points  in- 
dicate tenths,  but  it  can  be  read  by  estimation  to  hun- 
dredths. It  is  possible  to  make  the  reading  much  more 


*  Engineering  News,  Vol.  8,  p.  415. 
f  Engineering  News,  Vol.  8,  p.  289. 


ART.  2]  LEVELING    RODS. 


accurate  by  dividing  the  oblique  side  of  the  triangle, 
than  can  be  done  when  the  rod  is  graduated  according  to 
the  principle  of  Fig.  74. 

Any  pattern  suitable  for  a  stadia  rod  (see  Figs.  49-52, 
page  180)  can  be  used  in  making  a  self-reading  leveling 
rod.  The  pattern  may  be  painted  or  stenciled  directly 
upon  the  wood,  or  it  may  first  be  drawn  or  painted 
upon  paper,  and  then  fastened  on  the  rod  with  varnish 
or  any  glue  not  soluble  in  water.  Strips  of  cloth  or 
paper  containing  scales  for  this  purpose  are  sold  by 
dealers  in  engineering  stationery.  For  a  few  points 
applicable  to  the  manufacture  of  home-made  self-read- 
ing level  rods,  see  §§  208  and  209. 

271.  Target  vs.  Self-reading  Rods.  Probably  the 
former  are  the  more  common  now,  but  as  the  advan- 
tages of  the  latter  are  becoming  better  understood 
they  are  being  more  generally  used.  The  chief  advan- 
tage of  self-reading  rods  is  the  saving  of  time.  Setting 
the  target  to  some  exact  point  in  accordance  with 
directions  given  from  a  distance,  is  a  tedious  process  at 
best.  After  a  little  familiarity  with  the  pattern  of  the 
self-reading  rod,  the  height  of  the  line  of  sight  upon 
the  rod  can  be  read  very  quickly. 

On  the  other  hand,  target  rods  must  of  necessity  be 
capable  of  greater  precision  than  self-reading  ones,  but 
the  difference  in  accuracy  is  not  so  great  as  might  at 
first  seem.  The  accuracy  of  leveling  depends  upon  a 
number  of  things  (§§  310-19),  of  which  the  reading  of 
the  position  of  the  line  of  sight  upon  the  rod  is  one  of 
the  least  important,  and  all  of  the  others  are  independ- 
ent of  the  kind  of  rod.  Reading  the  position  of  the 
target  to  thousandths  of  a  foot  is  unnecessary  and  use- 
less, unless  all  other  parts  of  the  work  are  equally  pre- 
cise. The  thing  to  be  sought  is  proportionate  accuracy 
in  all  parts  of  the  work.  If  several  independent  read- 
ings of  a  rod  be  made  upon  the  same  point,  the  differ- 


236  SPIRIT-LEVELS.  [CHAP.  XI 

ence  between  the  various  readings  will  probably  be 
considerably  larger  than  the  probable  error  of  reading 
a  self-reading  rod  (see  Tables  I  and  II,  Appendix  III). 

A  self-reading  rod  is  generally  used  in  precise  level- 
ing, two  or  three  hairs,  usually  the  latter,  being 
read  to  reduce  the  error  of  reading.  Three  observa- 
tions on  a  self-reading  rod  are  probably  more  accurate 
than  a  single  observation  upon  an  ordinary  target,  and 
can  be  made  in  about  the  same  time.  The  difference 
between  the  readings  of  the  central  and  the  two  side 
hairs  affords  a  check  on  the  reading  and  also  on  the 
length  of  sight. 


ART.  3.     TESTING  THE  LEVEL. 

272.  THE  BUBBLE  TUBE.  Its  Form.  A  bubble  tube 
is  a  glass  tube  bent  or  ground  so  that  its  inside 
upper  surface  is  circular  on  a  longitudinal  section. 
Since  the  position  of  the  bubble  is  determined  by  read- 
ing the  position  of  its  ends,  a  longitudinal  section  of 
the  interior  should  be  of  uniform  curvature,  i.e.,  cir- 
cular, and  the  tube  should  not  be  in  the  least  conical. 

These  conditions  are  satisfied  if  both  ends  of  the 
bubble  move  over  equal  spaces  for  equal  displacements 
of  the  bubble  tube  in  altitude.  This  condition  is  not 
an  absolute  necessity,  since  the  bubble  should  always 
stand  in  the  middle  when  an  observation  is  made.  But 
in  very  delicate  instruments  it  is  nearly  impossible  to 
keep  the  bubble  in  the  middle  ;  and  hence,  if  the  above 
condition  is  satisfied,  the  bubble  need  not  be  brought 
exactly  to  the  center  each  time,  for  its  position  may  be 
noted  and  a  correction  applied. 

To  test  the  uniformity  of  the  curvature  of  the  tube, 
sight  at  the  target  of  a  leveling  rod,  and  read  the  posi- 
tion of  the  bubble;  and  then  move  the  target  over  equal 


ART.  3]  TESTING    THE    LEVEL.  237 

spaces,  sighting  upon  it  and  noting  the  movement  of 
both  ends  of  the  bubble  for  each  position. 

Or  this  test  may  be  made  without  the  aid  of  a  tele- 
scope, by  fastening  the  bubble  tube  to  a  board  hinged 
so  as  to  move  in  a  vertical  plane,  and  measuring  the 
tangent  of  the  angle  of  elevation.  A  micrometer 
screw*  affords  the  easiest  and  best  means  of  measur- 
ing the  tangent.  Such  a  contrivance  is  known  as  a 
level-trier. 

A  less  delicate  method  of  testing  the  above  condition 
is  to  note  whether  the  bubble  expands  and  contracts 
equally  both  ways  from  the  center  during  changes  of 
temperature.  In  making  these  observations  great  care 
must  be  taken  that  the  inclination  of  the  level  tube  is 
not  changed  by  external  forces  or  by  a  change  of  temper- 
ature of  the  parts  of  the  instrument. 

273.  Its  Sensitiveness.  The  most  important  condition 
to  be  fulfilled  by  the  level  tube  is  that  the  bubble  shall 
be  sensitive.  The  sensitiveness  depends  upon  the  radius 
of  curvature,  or,  in  other  words,  upon  the  distance  the 
bubble  moves  for  any  change  of  inclination. 

To  measure  the  sensitiveness,  proceed  as  follows: 
Bring  the  bubble  nearly  to  the  center,  and  sight  upon  a 
rod  held  vertically.  Raise  or  lower  one  end  of  the  level, 
by  operating  the  foot  screws,  until  the  bubble  moves 
about  as  far  to  the  other  side  of  the  center,  and  sight 
at  the  rod  again.  Let  h  —  the  difference  of  rod  read- 
ings; D  —  the  distance  from  the  instrument  to  the  rod; 
m  =  the  distance  the  bubble  moved;  d  —  the  length  of 
one  division  of  the  scale;  n  —  the  number  of  divisions 
the  bubble  moved;  I  —  the  change  of  inclination  of  the 
line  of  sight;  R  —  radius  of  curvature  of  the  level  tube; 


*  A  fair  micrometer  screw  can  be  made  of  a  tangent  screw  from  a  transit 
or  leveling  instrument,  to  which  is  attached  a  graduated  cardboard  disk. 


238 


SPIRIT-LEVELS. 


[CHAP,  xi 


and  F=  the  angular  value  of  one  division  of  the  scale. 


FIG.  76. 


Then  in  Fig.  76,  from  the  approximately  similar  tri- 
angles ABC  and  OPQ,we  have  : 


(0 


tan  7= 


Hence 


tan  \"nD       0.00000485 nD' 


(3) 


The  sensitiveness  of  a  bubble  is  generally  stated  by 
giving  the  angle  corresponding  to  a  movement  of  one 
inch.  Good  engineer's  levels  have  a  motion  of  200  to 
120"  per  inch,  or  a  radius  of  curvature  of  85  to  140  feet. 
Levels  of  precision  have  a  motion  of  about  50  to  30" 
per  inch,  or  a  radius  of  curvature  of  340  to  600  feet. 

274.  SENSITIVENESS  vs.  MAGNIFYING  POWEK.  The  mag- 
nifying power  of  the  telescope  and  the  sensitiveness  of 
the  level  should  be  so  proportioned  to  each  other  that 

*  The  engineer  should  carry  the  value  of  tan  l''  in  his  mind,  as  it  is  of 
frequent  application.  To  assist  the  memory  notice  that  the  value  is  approx- 
imately 5  zeros  and  a  5. 


ART.   3]  TESTING    THE    LEVEL.  239 

the  least  perceptible  motion  of  the  bubble  will  cause 
sufficient  motion  of  the  cross  hair  on  the  rod  to  be 
easily  noticed;  and  vice  versa,  the  least  noticeable  motion 
of  the  cross  hair  on  the  rod  should  cause  a  perceptible 
movement  of  the  bubble.  A  higher  power  or  a  more 
sensitive  level  than  that  required  by  the  above  con- 
dition adds  nothing  to  the  accuracy  of  the  instrument 
and  is  even  worse  than  useless,  for  the  former  causes  a 
loss  of  brilliancy  of  the  object  and  the  latter  an  annoy- 
ance in  leveling  the  instrument.  Of  two  levels  in  the 
writer's  possession,  one  had  a  magnification  of  17  and 
a  radius  of  the  bubble  of  84  feet,  and  th'e'bther  a  mag- 
nification of  27  and  a  radius  of  22  feet.  The  simple 
expedient  of  changing  the  bubbles  improved  both 
instruments  very  much.  A  third  instrument  is  inferior 
to  the  other  two  in  definition,  although  it  has  a  bubble 
of  165  feet  radius. 

275.  TELESCOPE  SLIDE.  The  telescope  slide  should  be 
straight  and  the  optical  center  should  move  in  the  line 
of  collimation.  The  wye  level,  after  having  been  colli- 
mated^  278),  may  be  tested  by  the  method  of  paragraph 
i,  §  121.  The  slide  is  first  tested  for  deviation  from  a 
vertical  plane  by  setting  a  row  of  points  with  the  tele- 
scope in  a  chosen  position,  and  then  revolving  it  180° 
in  the  wyes  and  setting  a  second  row.  An  easy  method 
of  accomplishing  this  is  to  set  the  instrument  in  a  cut 
or  low  place,  and  sight  at  a  plumb-line,  marking  the 
point  on  the  head  of  a  stake.  Care  must  be  taken  not 
to  disturb  the  instrument  when  moving  the  slide.  If 
there  is  any  play  or  looseness  in  the  object-glass  slide 
it  will  be  almost  impossible  to  make  this  test  satisfac- 
torily. Having  completed  this  test,  revolve  the  tele- 
scope 90°  from  its  first  position  and  repeat  as  above.  If 
the  two  rows  of  points  do  not  coincide,  either  the  slide 
is  not  straight  or  it  does  not  move  in  the  right  direction. 
If  the  direction  of  motion  of  the  slide  is  adjustable,  the 


240  SPIRIT    LEVELS.  [CHAP.   XI 

engineer  can  determine  the  source  of  error  only  by 
trial.  In  doing  this  remember  that  a  movement  of  the 
back  end  of  the  objective  slide  disturbs  the  adjustment 
of  the  line  of  collimation.  A  better  method  of  testing  the 
slide  of  the  wye  level  will  be  given  when  considering 
the  method  of  collimating  that  instrument  (see  §  280.) 

The  slide  of  the  dumpy  level  can  not  be  tested  by  the 
method  of  paragraph  2,  §  121,  since  the  telescope  can  not 
be  reversed  about  its  own  axis.  It  may  be  tested,  after 
having  been  collimated  (§  285),  by  taking  readings  on  a  row 
of  stakes  as  described  in  paragraph  2,  §  121,  and  then 
moving  the  instrument  to  the  other  end  of  the  row  and 
sighting  upon  them  again.  If  the  differences  of  level 
relative  to  the  stake  first  sighted  at  are  the  same  both 
times,  the  slide  is  straight.  This  is  not  a  very  good  test 
owing  to  the  errors  of  observation,  but  it  is  the  only 
one  available. 

276.  RINGS   AND   WYES   OF   THE   WYE  LEVEL.    The 

rings  should  be  cylinders  of  the  same  size,  and  the  wyes 
should  present  the  same  angle  in  the  direction  of  the 
telescope  tube.  The  last  is  not  likely  to  be  in  error  an 
appreciable  amount.  The  wyes  may  be  compared  by 
taking  them  off  and  placing  them  side  by  side,  or  by 
carefully  marking  the  angle  of  each  upon  a  piece  of 
paper. 

The  equality  of  the  rings  is  a  very  important  matter, 
but  is  often  overlooked.  Neglecting  to  examine  this 
point  makes  the  accuracy  of  the  engineer's  work  depend 
upon  that  of  an  unknown  workman. 

The  size  of  the  rings  can  not  be  tested  by  any  system 
of  reversals,  and  can  be  examined  only  by  means  of 
some  auxiliary  instrument.  They  may  be  compared 
by  calipering  or  by  means  of  a  delicate  striding  level. 
The  student  will  do  well  to  inquire  into  the  accuracy  of 
these  methods.  But,  for  the  engineer,  the  best  method 
is  to  take  a  test  level,  as  will  be  described  in  §  284. 


ART.  4]  ADJUSTMENTS    OF    WYE    LEVEL.  241 

The  test  level  should  not  be  taken  until  all  the  adjust- 
ments have  been  made. 

If  the  rings  are  not  the  same  size,  the  instrument  is, 
in  effect,  a  dumpy  level,  and  must  be  adjusted  and  used 
as  such, — that  is  to  say,  the  telescope  must  not  be 
reversed  in  the  wyes  end  for  end,  as  in  the  wye  level. 


ART.  4.     ADJUSTMENTS  OF  THE  WYE  LEVEL.* 

277.  LEVEL  TUBE.  The  tangent  at  the  middle  of  the 
level  tube  (the  zero  of  the  scale)  should  be  parallel  to 
the  bottom  of  the  wyes.  The  adjustment  is  made  in 
two  steps:  the  first  is  to  bring  the  tangent  and  the  axis 
of  the  telescope  into  the  same  plane,  and  the  second  is 
to  bring  the  level  parallel  to  the  bottom  of  the  wyes. 

i.  In  making  this  adjustment,  it  is  best  to  have  the 
sun-shade  on  and  have  the  object-glass  run  out  for  the 
mean  length  of  sight,  for  then  the  weight  will  be 
symmetrical  about  the  vertical  axis  and  the  level  will 
not  be  affected  by  any  unequal  strain. 

Clamp  the  vertical  axis  of  the  instrument,  loosen  the 
clips  that  hold  the  telescope  in  the  wyes,  rotate  the  tele- 
scope in  the  wyes  until  the  level  is  about  vertically 
under  the  telescope,  and  bring  the  bubble  to  the  middle 
by  moving  the  foot  screws.  Then  rotate  the  telescope 
in  the  wyes,  so  that  the  level  tube  swings  a  few  degrees 
to  one  side  of  a  vertical  plane.  If  the  bubble  changes 
its  position  longitudinally,  it  shows  that  the  axis  of  the 
telescope  and  the  level  are  not  in  the  same  plane.  Cor- 
rect all  the  error  by  moving  the  azimuth  screws  of  the 
level. 

After  having  made  the  bubble  tube  at  least  approxi- 
mately parallel  to  the  bottoms  of  the  rings,  if  the  bubble 
runs  toward  the  same  end  when  swung  on  both  sides 

*  For  general  remarks  on  adjustments,  see  §  37. 


242  SPIRIT    LEVELS.  [CHAP.  XI 

of  the  vertical,  the  tube  is  conical  (see  §  272).  Correct 
work  can  not  be  done  with  a  tube  of  this  form,  owing 
to  the  effect  of  changes  of  temperature  and  to  the  un- 
certainty of  the  level's  being  vertically  under  the  tele- 
scope. 

2.  Having  made  the  first  step,  as  above,  bring  the 
bubble  to  the  middle,  and  carefully  reverse  the  telescope 
in  the  wyes,  end  for  end.  If  the  bubble  does  not  stand 
in  the  middle  after  reversal,  correct  half  the  difference 
by  the  altitude  screws  of  the  level,  and  the  other  half 
by  the  foot  screws.  Notice  that  if  the  angles  of 
the  wyes  are  the  same  and  symmetrically  placed,  this 
brings  the  level  parallel  to  the  bottom  of  the  wyes, 
whether  the  rings  are  of  the  same  size  or  not. 

If  the  rings  are  of  the  same  size,  this  adjustment 
brings  the  tangent  of  the  level  parallel  to  the  axis  of 
the  rings.  The  equality  of  the  rings  must  be  tested  by 
a  test  level  (§  284). 

278.  CROSS  HAIRS.  The  line  of  collimation  should 
coincide  with  the  line  of  the  centers  of  the  rings.  To 
make  this  adjustment,  /.<?.,  to  collimate  the  instrument, 
focus  upon  some  well-defined  point  (§  94)  200  or  300 
feet  distant,  and  turn  the  telescope  over  in  the  wyes. 
If  the  intersection  of  the  cross  hairs  has  moved  from 
the  point  sighted  at,  bring  it  half-way  back  by  moving 
the  screws  in  the  diaphragm  which  carries  the  cross 
hairs  (B  B,  Fig.  62,  page  222),  and  correct  the  other  half 
by  the  foot  screws  and  the  tangent  screw  to  the  vertical 
axis. 

Since  the  cross  hairs  may  not  have  been  moved  over 
exactly  half  the  difference,  and  since  the  instrument 
may  have  been  disturbed,  the  adjustment  should  be 
repeated.  The  instrument  need  not  be  level  while 
making  this  adjustment.* 

*  The  next  two  sections  relate  to  the  testing  of  the  direction  of  motion  of 
the  object-glass  slide.      If   the  instrument  is   non-adjustable  and   has  been 


ART.  4]  ADJUSTMENTS    OF    WYE    LEVEL.  243 

279.  Since   it  is   tedious  to   always  bring  the   target 
exactly  to  the  middle  of  the  field  of  view,  the  horizontal 
hair  should   be   as   nearly  horizontal  as  possible;  and 
therefore,  after  having  adjusted  the  line  of  collimation, 
bring  the  vertical  axis  vertical,  close  the  clips  at  the  top 
of  the  wyes  (or  bring  the  mark  on  the  ring  to  coincide 
with  the  mark  on    the  wye — §  259),  and  bisect  the  tar- 
get  of  a  leveling  rod  with  the  horizontal  hair.     Move 
the  telescope  in  azimuth,  keeping  it  level,  and   then  if 
the  target  is  not  bisected  throughout  the  length  of  the 
hair,  rotate  the  reticule  until  it  is.     The  horizontal  hair 
will  now  be  horizontal,  and  the  other  one  will  be  vertical 
or  nearly  so,  since  the  two   are  supposed  to  have  been 
placed  perpendicular  to  each   other.     It  will  be  at  least 
nearly  enough  vertical  to  plumb  the  rod  by. 

Instead  of  adjusting  the  horizontal  hair  as  above,  it 
is  customary  to  place  the  vertical  hair  vertical  by  sight- 
ing at  a  corner  of  a  building.  Since  the  rod  may  not 
be  brought  exactly  to  the  center  of  the  field,  it  is  more 
important  that  the  horizontal  hair  should  be  truly  hori- 
zontal than  that  the  vertical  one  should  be  vertical. 
For  this  reason,  it  is  better  to  adjust  the  horizontal  thaia 
the  vertical  hair. 

280.  When  the   instrument  has  been  collimated    for 
the  above  distance,  focus  upon  a  point  very  near  the 
instrument,  and  test  the  adjustment  of  the  line  of  col- 
limation for  this  distance.     If  the  instrument  is  not  in 
adjustment    for    the    second    point,    either    the    optical 
center  is  not  in  the  axis  of    the  rings,  or    the  line    of 
motion  is   not   parallel  to   the  axis   of   the   rings.     The 
engineer  has   no  means  of   determining  which   of   these 
conditions  causes  the  error,  except  by  trial.     The  former 
may  be  caused  by  the  optical  center's  not   lying   in  the 

tested  in  this  respect,  the  operations  described  in  the  next  two  sections  may  be 
omitted  in  making  the  adjustments.  If  the  direction  of  motion  is  adjustable, 
the  engineer  should  occasionally  test  it. 


244  SPIRIT    LEVELS.  [CHAP.   XI 

center  of  figure  of  the  lens,  or  by  the  telescope  slides 
not  being  straight.  With  an  instrument  which  is  pro- 
vided with  a  means  of  adjusting  the  direction  of 
motion  of  the  object-glass  slide  (C,  Fig.  62,  page  222), 
the  engineer  may  move  the  inner  end  of  the  slide  until 
half  the  error  is  corrected;  and  then  collimate  again 
upon  the  first  point,  and  test  the  adjustment  upon  the 
second.  If  the  instrument  is  in  adjustment  for  the  two 
points  at  the  same  time,  it  is  probable  that  the  optical 
center  lies  in  the  axis  of  the  rings  and  that  the  direction 
of  motion  coincides  with  that  line. 

281.  It  is  generally  held  that  if  an  instrument  is  in 
collimation  for  two  different  distances  at  the  same  time, 
the  line  of  sight  must  be  in  the  axis  of  the  rings;  but 
this  is  not  necessarily  true.  In  general,  if  either  the 
intersection  of  the  cross  hairs  or  the  optical  center  of 
the  objective  does  not  lie  in  the  axis  of  the  rings,  the 
line  of  collimation  will  describe  the  surface  of  a  cone 
when  the  telescope  is  revolved  in  the  wyes;  and  for  any 
given  position  of  the  optical  center,  it  is  possible  to 
have  such  a  position  of  the  cross  hairs  and  correspond- 
ing motion  of  the  slide  that  the  instrument  may  be  in 
collimation  for  two  points  at  the  same  time  and  still 
the  line  of  collimation  not  be  in  the  axis  of  the  rings. 
That  is  to  say,  if  the  instrument  is  not  in  collimation, 
the  line  of  sight  will  describe  the  surface  of  a  variable 
cone,  the  points  collimated  upon  being  two  positions  of 
the  vertex.  Such  a  relation  of  parts  could  be  obtained 
only  by  a  series  of  successive  approximations.  The 
only  check  against  the  occurrence  of  this  condition  is  a 
test  level  (§  284). 

Another  particular  case  is  that  in  which  the  line  of 
collimation  describes  a  cylinder.  This  occurs  when  the 
optical  center  and  the  intersection  of  the  cross  hairs  are 
on  the  same  side  of,  and  equally  distant  from,  the  axis 
o£  the  rings.  This  state  of  affairs  will  be  revealed  by 


ART.  4]  ADJUSTMENTS    OF    WYE    LEVEL.  245 

the  instrument's  being  out  of  collimation  the  same 
amount  for  all  distances.  In  this  case  the  line  of  colli- 
mation will  be  parallel  to  the  level  (assuming  the  rings  to 
have  been  found  to  be  of  the  same  size),  and  therefore 
the  instrument  is  in  adjustment.  If  the  optical  center 
does  not  lie  in  the  line  of  the  centers  of  the  rings,  the 
cross  hairs  maybe  moved, by  a  series  of  trials,  until  the 
line  of  sight  shall  describe  the  surface  of  a  cylinder. 
Notice,  however,  that  this  can  occur  only  in  those  in- 
struments in  which  the  line  of  motion  is  parallel  to  the 
axis  of  the  rings. 

282.  CENTERING   THE   EYE-PIECE.      If   after   having 

brought  the  intersection  of  the  cross  hairs  into  the  axis 
of  the  ring,  the  same  field  of  view  is  not  presented  dur- 
ing an  entire  revolution  of  the  telescope  in  the  wyes, 
the  fault  is  in  the  centering  of  the  eye-piece;  in  other 
words,  the  optical  axis  of  the  eye-piece  does  not  coincide 
with  that  of  the  objective.  Some  instruments  are  pro- 
vided with  an  arrangement  for  correcting  this  by  a 
motion  of  the  inner  end  of  the  eye-piece, — see  A  A,  Fig. 
62,  page  222.  (The  adjusting  screws  are  covered  by  a 
ring  which  can  easily  be  slipped  off.)  Strictly,  this  ad- 
justment does  not  place  the  two  axes  parallel;  it  only 
makes  the  optical  axis  of  the  eye-piece  intersect  the 
optical  axis  of  the  objective  in  the  plane  of  the  image. 
However,  the  only  effect  is  a  slight  blurring  of  opposite 
sides  of  the  image. 

Inverting  telescopes  have  no  such  adjustment;  and 
it  would  be  an  advantage  if  it  were  omitted  in  telescopes 
with  an  erecting  eye-piece;  but  this  can  not  be  done, 
owing  to  mechanical  reasons. 

283.  WYES.     The    line  of   the   bottoms  of  the  wyes 
should  be  perpendicular  to  the  vertical  axis  of  the  in- 
strument.    When  the  instrument  is  in  adjustment,  the 
bubble    tube    is    parallel  to  the    bottom    of    the  wyes 
(§  277)>  and  therefore  this  adjustment  is  equivalent  to 


246  SPIRIT    LEVELS.  [CHAP.  XI 

placing  the  bubble  tube  perpendicular  to  the  vertical 
axis  of  the  instrument.  Notice  that  this  adjustment  is 
for  convenience  only,  as  it  in  no  wise  affects  the  accu- 
racy of  the  work,  but  merely  saves  the  labor  of  leveling 
up  every  time  the  telescope  is  moved  in  azimuth. 

To  make  this  adjustment  bring  the  bubble  to  the  mid- 
dle by  turning  the  foot  screws,  and  turn  the  telescope 
180°  in  azimuth.  If  the  bubble  does  not  stand  in  the 
middle  after  reversal,  correct  half  the  error  with  the 
nuts  on  the  lower  ends  of  the  wyes  (Fig.  62,  page  222), 
and  the  other  half  with  the  foot  screws.  Since  the  tele- 
scope may  not  have  been  revolved  exacfe-ly  180°  in  azi- 
muth, test  the  adjustment  for  a  position  90°  from  the  one 
used  as  above,  and  then  repeat  in  the  first  position. 

284.  TEST  LEVEL.  After  having  made  the  usual  ad- 
justments of  the  wye  level,  it  is  absolutely  necessary 
that  they  should  be  verified  by  taking  a  test  level. 

To  take  a  test  level,  drive  two  pegs  into  the  ground, 
say,  400  feet  apart,  set  the  instrument  exactly  half-way 
between  them,  and  carefully  determine  the  difference 
of  level.  A  line  joining  the  two  positions  of  the  target 
is  a  level  line,  and  the  difference  between  the  readings 
is  the  true  difference  of  level  however  much  the  instru- 
ment may  be  out  of  adjustment  (§  126).  Make  several 
determinations  of  the  difference  of  level  for  a  check. 
Next,  set  the  instrument  near  one  of  the  pegs,  say,  10 
feet  beyond  it,  and  at  least  nearly  in  line  with  the  other, 
and  re-determine  the  difference  of  level,  making  several 
pairs  of  observations. 

If  the  difference  is  the  same  both  times,  it  is  certain 
that  all  the  conditions  enumerated  in  the  preceding  dis- 
cussion have  been  satisfied.  That  is,  an  agreement  in 
the  test  level  proves  (i)  that  the  slide  is  straight,  (2) 
that  the  line  of  motion  is  parallel  to  the  line  of  centers 
of  the  rings,  (3)  that  the  rings  are  of  the  same  size,  (4) 
that  the  level  is  parallel  to  the  bottom  of  the  wyes,  and 


AkT.  5]  ADJUSTMENTS    OF    1>UMPY    LEVEL.  $tf 

(5)  that  the  line  of  collimation  is  parallel  to  the  line  of 
the  centers  of  the  rings. 

If  the  differences  for  the  two  positions  of  the  instru- 
ment are  not  the  same,  the  error  is  due  to  defect  in  one 
or  more  of  the  five  particulars  mentioned  above.  If  the 
slide  is  not  straight,  the  only  remedy  is  in  a  new  instru- 
ment. Some  instruments  are  provided  with  a  means  of 
changing  the  direction  of  motion  of  the  object-glass 
slide;  but  it  is  at  least  doubtful  whether  this  is  any  ad- 
vantage to  an  instrument.  When  no  means  of  adjust- 
ment is  provided,  the  only  remedy  is  a  new  instru- 
ment. If  the  rings  are  not  of  the  same  size,  the  instru- 
ment can  be  adjusted  and  used  as  a  dumpy  level.  If 
the  fourth  or  fifth  conditions  are  not  satisfied,  a  re- 
adjustment is  sufficient. 

ART.  5.     ADJUSTMENTS  OF  THE  DUMPY  LEVEL.* 

285.  COLLIMATION.  It  is  required  to  make  the  line  of 
collimation  and  the  level  parallel  to  each  other.  This 
may  be  accomplished  in  either  of  two  ways:  (i)  by  bring- 
ing the  line  of  collimation  parallel  to  the  level,  or  (2) 
by  bringing  the  level  parallel  to  the  line  of  collimation. 
In  case  both  the  level  and  the  line  of  collimation  are  mov-. 
able,  the  latter  method  is  to  be  preferred;  for  then  the 
intersection  of  the  cross  hairs  can  first  be  placed  in  the 
optical  axis  of  the  telescope,  and  not  be  disturbed  by 
this  adjustment.  The  best  form  of  instrument  is  that 
in  which  the  level  is  fixed,  for  then  the  maker  can  place 
the  level  parallel  to  the  optical  axis  before  he  fastens  the 
telescope  to  the  standards;  and  when  the  line  of  collima- 
tion is  adjusted,  it  will  coincide  with  the  optical  axis. 

i.  To  place  the  line  of  collimation  parallel  to  the  level, 
set  the  instrument  upon  a  level  piece  of  ground,  and 

*  For  general  remarks  on  adjustments,  see  §  37. 


248  SPIRIT   LEVELS.  [cHAP.  XI 

bring  the  bubble  to  the  middle  of  its  race,  read  the  rod 
upon  a  point,  say,  200  feet  distant,  turn  the  instrument 
about,  bring  the  bubble  to  the  middle,  and  sight  in  the 
opposite  direction  upon  a  point  at  exactly  the  same  dis- 
tance as  before.  The  difference  of  readings  is  the  true 
difference  of  level.  Make  several  determinations  of  the 
difference  for  a  check.  Then  move  the  instrument  near 
one  of  the  points,  say,  10  feet  beyond  it,  bring  the  bubble 
to  the  middle,  and  sight  upon  each  point.  If  the  second 
difference  is  not  equal  to  the  first,  correct  the  error  by 
moving  the  cross  hairs  over  a  space  on  the  farther  rod 
equal,  for  the  case  supposed  above,  to  f-^ths  of  the  ap- 
parent difference  of  level  (see  the  second  paragraph  of 
§  126).  When  the  difference  of  readings  is  equal  to  the 
true  difference  of  level,  the  line  of  collimation  is  horizon- 
tal and  therefore  parallel  to  the  level. 

2.  To  place  the  level  parallel  to  the  line  of  collimation, 
set  the  instrument  midway  between  two  points  and 
determine  the  true  difference  of  level.  Next  move  the 
instrument  near  one  of  the  points,  and  by  means  of  the 
foot  screws  change  the  inclination  of  the  telescope  until 
the  difference  of  the  readings  is  the  same  as  when  the 
instrument  was  in  the  middle  (see  §  126).  The  line  of 
^sightis  then  horizontal  (see  Fig.  29,  page  113).  Without 
altering  the  inclination  of  the  line  of  sight,  raise  or 
lower  one  end  of  the  level  until  the  bubble  is  in  the 
middle.  The  level  and  line  of  sight  are  then  parallel. 

Notice  that  this  adjustment  does  not  necessarily  make 
the  line  of  collimation  coincide  with  the  optical  axis  of 
the  telescope;  but  it  can  be  shown*  that  if  the  direction 
of  motion  of  the  slide  is  parallel  to  the  optical  axis,  the  instru- 
ment will  still  give  correct  results  when  adjusted  as 
above. 

286.  WYES.  For  convenience,  the  level  and  line  of 
collimation  should  be  perpendicular  to  the  vertical  axis. 

*  Rankine's  Civil  Engineering,  p.  84. 


ART.  6]        ADJUSTMENTS    OF    LEVEL    OF    PRECISION.  249 

To  make  this  adjustment,  bring  the  bubble  to  the  mid- 
dle by  means  of  the  foot  screws  alone,  and  revolve  the 
instrument  180°  in  azimuth.  If  the  bubble  is  in  the 
middle  after  reversal,  it  is  adjusted;  if  not,  correct  half 
the  error  by  raising  or  lowering  one  of  the  standards 
connecting  the  telescope  with  the  tripod  head,*  and  the 
other  half  by  the  foot  screws.  Repeat  the  operation 
until  the  bubble  does  not  move  in  the  tube  when  the 
instrument  is  turned  in  any  position  on  its  vertical  axis. 
In  some  instruments  there  are  no  means  of  adjusting 
the  height  of  the  standards  or  wyes,  in  which  case  it  be- 
comes necessary  to  make  the  level  perpendicular  to  the 
vertical  axis  before  adjusting  the  line  of  collimation, 
and  to  adjust  the  line  of  collimation  by  the  first  method. 

ART.  6.     ADJUSTMENTS  OF  LEVEL  OF  PRECISION. 

287.  As  the  adjustments  of  levels  of  precision  do  not 
commonly  differ  materially  from  those  of  the  ordinary 
forms,  the  method   need   not  be  re-stated    here.     It  is 
customary  to  adjust  the  instrument  as  carefully  as  pos- 
sible,   and    then    determine    the    error   of    adjustment. 
Each  observation  may  then  be  corrected,  thus  affording 
a  check  between  the  members  of  a  double  observation. 
The  tests  and  adjustments  of  the  form  shown  in  Fig.  64 
(page  225)  are  the  same  as  those  of    the  wye  level,  with 
the  following  tests  in  addition. 

288.  The   telescope  when   raised   or  lowered  by    the 
micrometer    screw    should    move    in    a   vertical    plane. 
This    adjustment    is    necessary,  since    it    is  not    always 
practicable  to  have  the  line  of  sight  exactly  horizontal  at 
the  moment  of  sighting.     To  test  this  adjustment,  care- 

*  The  screws  for  raising  or  lowering  the  standards  are  on  the  under  side 
of  the  cross-bar,  and  are  to  be  turned  with  a  screw-driver.  Owing  to  the  unex- 
posed  position  of  these  screws,  they  are  not  liable  to  be  turned  accidentally, 
which  is  one  of  the  advantages  of  this  form  of  level. 


SPIRIT    LEVELS.  [CHAP.  XI 


fully  level  the  instrument,  and  sight  upon  a  plumb-line; 
and  then  move  the  telescope  in  altitude  and  see  whether 
the  intersection  of  the  cross  hairs  follows  the  plumb- 
line.  If  it  does  not,  the  only  remedy  is  to  return  the 
instrument  to  the  maker. 

289.  The  angular  value   of  one  revolution   of  the  mi- 
crometer screw  should  be  known  approximately.     This 
can   be  determined   only  approximately,  owing  to  the 
eccentric  position  of  the  line  of  sight  with  reference  to 
the  pivots  on  which  the  telescope  turns.     The  value  of 
a  revolution  of  the  micrometer  is  useful  only  in  correct- 
ing an  observation  made  when    the  line  of  sight   is   not 
exactly  horizontal,  and  hence  an  approximate  value  is 
sufficient  for  any  case  arising  in  practice.     To  determine 
an  approximate  value  of  one  revolution,  bring  the  line 
of  sight  nearly   horizontal,   read   the   micrometer,  and 
sight  upon  a  level  target  at  a  distance  D.     Then  turn  the 
micrometer  screw  one  revolution,  and  read  the  rod  again. 
Representing  the  difference  of  rod   readings  by  d,  the 
angular  value   of  one   revolution   in   seconds  of  arc  is 

--  r/  .  To  reduce  the  error  of  observation,  deter- 
D  tan  i" 

mine  the  value  of  several  —  say,  three  or  four  —  revolu- 
tions and  divide  the  result  by  the  number  of  revolutions. 

290.  The  collars,  or  rings,  on   the   telescope,  /.<?.,  the 
bearings  in  the   wyes,  should   be   of  exactly  the   same 
diameter.     If  they  are  not,  the  inequality  must  be  deter- 
mined and  a  correction  computed.     The  inequality  of 
rings  is   the   only  instrumental   error   that   can   not   be 
eliminated  by  any  system   of  double   observations.     It 
may  be  eliminated  from  the  final  result  by  equal  back- 
and    fore-sights    (§326);  but   to    employ    the    check  of 
double  leveling   (§  306),  a  correction   for  inequality  of 
collars  must  be  applied  to  the  rod  reading. 

To  find  the  inequality  of   collars,  set  the   instrument 
firmly,  and  level  it  carefully.     Read   both   ends  of  the 


ART.  6]        ADJUSTMENTS   OF    LEVEL    OF    PRECISION.  351 

bubble,  and  take  the  mean;  reverse  the  bubble  tube 
end  for  end,  read  again,  arid  take  the  mean.  Half  the 
difference  of  these  means  is  the  inclination  of  the  top  of 
the  rings,  expressed  in  divisions  of  the  bubble  scale. 
Reverse  the  telescope  end  for  end  in  the  wyes,  and  re- 
peat the  entire  process  as  above.  To  meet  the  possibil- 
ity of  the  rings'  not  being  perfectly  round,  revolve  the 
telescope  about  the  optical  axis  and  determine  the  in- 
clination as  above  with  the  telescope  both  direct  and 
reversed.  If 

Ed  =  the  inclination  when  telescope  is  direct  and  bubble 

tube  erect, 
Er  =  the   inclination    when   telescope   is  reversed    and 

bubble  tube  erect, 
Id  =  the  inclination  when  telescope  is  direct  and  bubble 

tube  inverted, 
fr  =  the    inclination  when   telescope   is  reversed   and 

bubble  tube  inverted, 

V  =  the  value  of  one  division  of  the  bubble  scale, 
the  correction  to  be  applied  to  an  observed  reading  at 
a  distance  £>>  is  equal  to 


sn 


If  the  object-end  ring  is  too  large,  the  line  of  sight  will 
be  inclined  downward  when  the  tops  of  the  rings  are 
horizontal. 

291.  For  the  most  accurate  work,  it  is  necessary  to 
determine  the  absolute  value  of  the  unit  of  the  rod, 
and  also  to  test  the  uniformity  of  the  graduation. 
The  coefficient  of  expansion  of  the  rod  and  the  gradu- 
ation of  the  attached  thermometer  should  likewise  be 
tested,  as  also  the  adjustment  of  the  level  by  which  the 
rod  is  kept  vertical. 


SPIRIT    LEVELS.  JCHAP.  Xl 


ART.  7.     USING  THE  LEVEL. 

292.  A  level  line  is  a  line  parallel  to  the  surface  of 
still  water.     A  horizontal  line  is  a  straight  line  tangent 
to  a  level  line.     Level  and   horizontal  are  frequently 
used  as  meaning  the  same  thing,  and  many  times  there 
is  no   appreciable  difference   between    the   two   (§319, 
equation  3). 

The  level  is  used  (i)  to  find  how  much  one  point  is 
above  or  below  a  level  line  passing  through  another 
point,  (2)  to  obtain  a  profile  of  a  line,  and  (3)  to 
locate  contour  lines,  grade  lines,  boundaries  of  em- 
bankments and  excavations,  etc.  Whatever  the  ulti- 
mate purpose  of  the  work,  the  immediate  object  for 
any  setting  of  the  instrument  is  to  find  how  much  one 
point  is  higher  or  lower  than  another,  and  this  is  ascer- 
tained by  obtaining  a  horizontal  line,  and  measuring 
how  far  each  point  is  below  this  line.  The  line  of 
sight  of  a  properly  adjusted  leveling  instrument  is  a 
horizontal  line;  and  as  the  instrument  is  revolved  in 
azimuth,  this  line  marks  a  horizontal  plane.  The  ver- 
tical distance  of  any  point  below  this  plane  is  meas- 
ured by  the  leveling  rod. 

293.  DlFFEBENTIAL  LEVELING.      This  consists  in   find- 
ing how  much  one  point  is  above  or  below  a  level  line 
passing  through  another  point.     If  the  two  points  are 
not  too  far  apart,  either  horizontally  or  vertically,  the 
difference  of  level  may  be  found  by  setting  the  instru- 
ment between   the  two  and   taking  a  reading  of   the 
level  rod  on  each  point.     The  difference  of  the  read- 
ings will  be  the  difference  of  level  required. 

If  the  two  points  are  so  far  apart,  either  horizontally 
or  vertically,  that  the  difference  of  level  can  not  be 
found  by  a  single  setting  of  the  instrument,  establish 
one  or  more  intermediate  points,  and  determine  the 


A.RT.   7]  USING    THE    LEVEL.  25$ 

difference  of  level  between  successive  pairs  of  points. 
The  algebraic  sum  of  the  difference  of  level  between 
the  successive  pairs  of  points  is  the  required  difference 
of  level  ;  or,  in  other  words,  the  sum  of  the  rod-read- 
ings for  the  odd-numbered  sights  minus  the  sum  of  the 
readings  for  the  even-numbered  sights  is  equal  to  the 
algebraic  difference  of  level. 

If  the  elevation  of  the  first  point  is  given  with  refer- 
ence to  any  datum,  as,  for  example,  sea-level,  the  ele- 
vation of  the  first  point  plus  the  reading  of  the  rod  on 
that  point  gives  the  elevation  of  the  plane  of  sight  ; 
and  the  height  of  the  plane  of  sight  minus  the  reading 
of  the  rod  on  the  second  point  gives  the  elevation  of 
that  point  above  datum.  The  rod  reading  on  the 
known  point  is  called  the  back-sight,  and  the  reading 
on  the  point  whose  elevation  is  to  be  determined  is 
called  the  fore '-sight.  Notice  that  the  terms  back-sight 
and  fore-sight  have  no  reference  to  directions.  As 
shown  above,  the  back-sight  reading  is  essentially  posi- 
tive, and  the  fore-sight  is  essentially  negative  ;  and 
therefore  the  back-sight  is  sometimes  also  called  ^plus- 
sight  and  the  fore-sight  a  minus-sight.  The  intermediate 
points  established  between  two  remote  points  for  the 
purpose  of  determining  the  difference  of  level  of  the 
latter,  are  called  turning  points.  Bench  marks,  or  simply 
benches,  are  points  of  more  or  less  permanent  character 
whose  elevations  are  determined  and  recorded  for  fu- 
ture reference.  In  the  notes,  benches  are  indicated  by 
B.  M.  or  B,  with  a  subscript  to  show  the  number  of  the 
bench.  Thus  B^  means  the  third  bench  from  the  be- 
ginning of  the  line. 

294.  Field  Routine.  The  rod-man  holds  the  rod  on 
the  -starting  point.  He  should  stand  directly  behind 
the  rod,  and  hold  it  vertically.  The  instrument-man, 
or  leveler,  sets  up  the  instrument  100  to  300  feet  (§  325) 
from  the  end  of  the  line,  in  the  direction  in  which  the 


254  SPIRIT    LEVELS.  fcHAP.  Xl 

work  is  to  proceed.  It  is  not  necessary  that  the  instru- 
ment be  set  on  the  line,  unless  the  distance  run  is  also 
desired,  in  which  case  the  instrument  should  be  ap- 
proximately on  line.  Having  leveled  the  instrument, 
the  leveler  sights  upon  the  rod,  and,  if  it  is  a  self-read- 
ing rod,  notes  the  reading  and  writes  it  down  as  a 
" -f-  sight"  (§  293,  third  paragraph). 

If  a  target  rod  is  used,  the  leveler  signals  the  rod- 
man  the  direction  in  which  to  move  the  target.  There 
is  great  variety  in  the  manner  of  making  these  signals. 
The  important  thing  is  that  they  should  be  easily  seen 
and  readily  understood.  A  very  good  system  of  sig- 
nals is  that  in  which  the  arm  raised  above  the  shoulder 
indicates  that  the  target  should  be  moved  up,  the 
arm  coming  more  nearly  horizontal  as  the  target  ap- 
proaches the  desired  point ;  when  the  target  is  right, 
the  arm  is  swung  horizontally,  or  in  a  circle,  at  which 
signal  the  rod-man  clamps  the  target.  Similarly,  the 
arm  held  below  the  shoulder  indicates  that  the  target 
should  be  moved  down.  The  rod-man  should  place 
the  target  as  near  at  the  proper  place  as  he  can  by 
estimation,  without  waiting  for  signals  from  the  leveler. 

After  the  target  has  been  lined  in  by  the  leveler,the  rod- 
man  should  wave  the  rod  slightly  to  and  fro,  being  sure 
that  it  passes  through  the  vertical  position;  and  in  the 
mean  time  the  man  at  the  instrument  notices  whether  the 
target  rises  to  the  line  of  sight,  i.e.,  whether  the  highest 
position  of  the  target  coincides  with  the  horizontal  hair. 
If  it  does  not,  the  leveler  directs  the  rod-man  to  move 
the  target.  The  instrument-man  can  tell  whether  the 
rod  leans  to  the  right  or  left,  by  comparing  it  with  the 
vertical  hair.  If  the  rod  is  not  vertical,  he  should  hold 
one  arm  up  vertically,  stand  where  the  rod-man  can  see 
him.  and  incline  his  body  in  the  direction  the  rod  should 
be  moved.  A  good  rod-man  will  seldom,  or  never,  need 
this  signal. 


ART.  7]  USING    THE    LEVEL.  255 

When  the  position  of  the  target  is  satisfactory,  the 
rod-man  calls  out,  first,  the  number  of  the  station  or 
stake,  which  the  leveler  calls  back  as  evidence  that  he 
understands;  and  then  the  rod-man  calls  out  the  read- 
ing, one  figure  at  a  time,  which  is  repeated  by  the  leveler 
as  he  records  it  in  his  note-book.  These  numbers  may 
be  read  and  called  off  while  the  rod-man  is  walking  to 
the  next  station. 

Having  obtained  the  reading  at  the  first  station,  the 
rod-man  proceeds  as  far  on  the  other  side  of  the  instru- 
ment as  the  instrument  was  from  the  first  station  (see 
§  326),  and  establishes  a  turning  point  by  holding  the 
rod  upon  some  fixed  point  or  upon  a  stake  driven  for 
the  purpose.  An  iron  pin  i  inch  in  diameter  at  the 
top  and  6  or  8  inches  long,  tapering  to  a  point,  makes  a 
good  turning  point;  or  a  thin  triangular  plate  of  iron, 
having  sides  about  8  to  12  inches  long  with  about  2 
inches  of  the  corners  turned  down,  and  having  a  small 
button  attached  to  the  middle  of  its  upper  surface, 
makes  an  excellent  turning  point  for  hard  ground  where 
there  is  no  grass,  weeds,  etc.  Whether  the  pin  or  plate 
is  employed,  a  suitable  handle  should  be  attached  to  it. 

A  reading  is  taken  upon  the  turning  point,  and  the 
result  is  recorded  as  a  "  —  sight."  The  instrument  is 
then  moved  beyond  the  turning  point,  and  set  up  again, 
a  second  reading  is  taken  upon  the  turning  point,  and 
the  result  is  recorded  as  a  "  -f-  sight."  Thus  the  work 
proceeds,  the  rod  and  the  instrument  alternately  being 
ahead. 

295.  The  Record.  The  record  for  differential  leveling 
is  shown  in  Table  V,  page  256.  The  form  is  so  simple 
that  it  does  not  need  much  explanation.  The  sum  of  the 
back-sights  minus  the  sum  of  the  fore-sights  gives  the 
algebraic  difference  of  elevation.  The  length  of  sight, 
which  may  be  obtained  by  the  principle  of  the  stadia 
(pp.  173-208),  is  not  required  except  to  check  the  equality 


256 


SPIRIT    LEVELS. 


[CHAP,  xi 


of  the  length  of  the  back-sight  and  the  fore-sight  (§  326), 
or  to  determine  the  distance  run. 

TABLE  V. 

FORM  OF  RECORD  FOR  DIFFERENTIAL  LEVELING. 


Sta. 

Length 
of 
Sight. 

+  Sights. 

—  Sights. 

Elevation 
of 
Benches. 

Remarks. 

£i 

I 

2 

B* 

3 
4 

#3 

3.214 
3-6I7 
5-23I 

2.431 
4-733 
3-214 

O.OOO 
+  1-879 

-1.506 

Corner   of   water-table 
at   N.-W.   corner  of 
court-house. 
Marked    point    on    W. 
abutment   of    White 
River  bridge. 

3i8c 

-f-  12.062 
—  10.383 

-  10.383 

-f  1.879 
4.215 
2.831 

3-217 

5-272 
3.821 

8.925 

—  12.310 

+    8.925 

•  J°0 

+  1-879 

-     3-385 

296.  PROFILE  LEVELING.    The  object  of  this  form  of 

leveling  is  to  obtain  a  profile  of  the  surface  along  an 
established  line.  Hence  it  is  necessary  to  determine  both 
the  horizontal  and  the  vertical  distances  from  the  initial 
point  to  the  points  at  which  rod  readings  are  taken. 
When  the  line  is  established,  stakes  are  driven  at  regu- 
lar intervals,  usually  100  feet  apart.  The  stakes  are 
numbered  in  order, — the  first  being  o,  so  that  the  num- 
ber of  any  stake  will  indicate  its  distance  from  the  be- 
ginning. The  leveler  is  to  obtain  the  elevation  of  the 
ground  at  each  of  these  stakes  and  at  as  many  interme- 
diate points  as  may  be  necessary  to  enable  him  to  draw 
a  fairly  accurate  profile.  The  loo-foot  stakes  are  called 
Stations,  and  the  points  between  the  i<?o-foot  stakes  are 


ART.   7]  USING    THE    LEVEL. 


called  pluses.  Thus  the  tenth  zoo-foot  stake  is  called  sta- 
tion 10,  or  simply  10;  and  an  intermediate  point  20  feet 
beyond  station  10  is  referred  to  as  station  io-j-  20. 

297.  Field  Routine.  The  field  work  is  very  much  like 
that  for  differential  leveling  (see  §  293-94).  The  rod  is 
first  held  on  the  ground  at  the  foot  of  the  first  stake, 
i.e.,  at  station  o,  or  on  a  bench  near  the  beginning  of  the 
line.  The  number  or  designation  of  the  point  upon 
which  the  rod  is  held  is  entered  in  the  first  column  of 
the  record  (see  Table  VI,  page  261);  and  a  brief  descrip- 
tion of  the  location  of  the  point  is  recorded  in  the  last 
column.  The  reading  itself,  which  obviously  is  a  -f- 
sight,  is  recorded  in  the  second  column. 

A  rod  reading  is  then  taken  at  each  of  the  stations  in 
order,  the  readings  being  recorded  in  the  —  S.  column 
under  the  sub-head  intermediates.  The  work  thus 
proceeds  until  a  point  is  reached  which  is  about  as  far 
from  the  instrument  as  the  point  of  beginning.  If  there 
is  no  firm  point  upon  which  to  place  the  rod,  the  rod- 
man  drives  a  level-peg  (§  294,  fifth  paragraph),  which 
he  carries  for  the  purpose,  until  the  top  of  it  is  nearly 
level  with  the  surface  of  the  ground,  and  holds  the  rod 
upon  it.  This  is  a  turning  point,  and  the  reading  is 
recorded  in  the  —  S.  column  under  the  sub-head  P.  S. 
If  the  turning  point  is  a  regular  station,  it  needs  no 
other  designation  in  the  station  column  ;  if  it  is  not  a 
regular  station,  but  is  in  the  line,  then  it  can  be  desig- 
nated as  a  plus  ;  and  if  it  is  not  in  the  line,  it  is  desig- 
nated as  peg  or  T.  P. 

The  instrument  is  next  moved  forward,  and  a  sight 
is  taken  upon  the  turning  point  that  the  height  of  the 
new  line  of  sight  may  be  determined  with  reference  to 
tne  first  station.  The  work  then  proceeds  from  the 
nirning  point  as  from  the  beginning.  The  first  sight 
taken  each  time  after  setting  up  the  instrument  is  a 
back-sight,  or  -j-  sight;  the  last  sight  taken  before  re- 


258  SPIRIT    LEVELS.  [CHAP.  XV 

moving  the  instrument  is  a  fore-sight,  or  —  sight;  and 
ail  others  are  intermediate  sights,  or  simply  interme- 
diates. 

The  back-sights  and  fore-sights  require  the  greatest 
care,  since  any  error  in  them  affects  all  subsequent  work, 
while  an  error  in  an  intermediate  affects  only  the  height 
of  that  single  point.  When  the  rod  is  set  upon  the 
ground,  as  is  usually  the  case,  and  a  target  rod  (§  264) 
is  used,  it  is  sufficient  to  read  on  turning  points  to  hun- 
dredths  of  a  foot  and  on  intermediates  to  the  nearest 
tenth;  although  frequently,  and  generally  improperly, 
the  rod  is  read  on  turning  points  to  thousandths  and 
on  intermediates  at  least  to  hundredths.  With  the 
Philadelphia  rod  (§  267),  it  is  rarely  necessary  to  use  the 
target  except  on  turning  points. 

The  rod-man  is  to  place  the  rod  at  the  foot  of  each 
stake,  and  at  any  point  in  the  line  where  there  is  any 
considerable  change  of  elevation.  He  is  to  determine 
the  position  of  such  points  by  stepping  from  the  last 
station,  and  this  distance  is  recorded  in  the  first  column 
of  the  record  (see  Table  VI,  page  261).  For  example, 
a  reading  having  been  taken  on  a  point  50  feet  from 
station  8,  the  record  is  made  as  in  Table  VI, — see  the 
line  under  the  record  for  station  8. 

298.  The  Record.  A  great  variety  of  forms  of  record- 
ing the  notes  have  been  proposed.  The  form  shown  in 
Table  VI  is  frequently  used,  but  a  slight  modification 
of  it  (see  §  300,  second  paragraph)  is  by  far  the  most 
common  in  this  country.  The  common  form  and  the 
forms  shown  in  Tables  VI  and  VII,  pages  261  and 
263,  are  examples  of  the  method  by  height  of  instrument. 
The  form  shown  in  Table  VIII,  page  264,  is  an  example 
of  the  method  by  differences.  The  last  is  not  much  used 
in  this  country,  but  apparently  it  is  the  favorite  with 
British  engineers, 


ART.   7]  USING    THE    LEVEL.  259 

299.  Modified  Method  by  Height  of  Instrument.      The 
first,,  second,   fourth,   and    fifth    columns   of   Table  VI, 
page  261,  have  already  been  explained  (§  297).      Since 
the  most  convenient  method  of  expressing  the  relative 
heights  of  a  series  of  points  is  to  refer  them  to  some 
common  datum,  and  since  it  is  immaterial  where  this 
reference  plane  is  chosen;  therefore,  if  the  elevation  of 
the  first  stake — station  o  or  B^ — has   not    been    deter- 
mined by  previous  operations,  it  is  customary  to  assume 
it  to  be  10,  100,  or  1,000  feet;  that  is  to  say,  the  datum 
is  assumed  at  this  distance  below  the  first  station.     The 
elevation  of  the  first  station  is  assumed  at  some  number 
supposed  to  be  greater  than  any  depression  on  the  line 
to  be  leveled,  so    as  to    obviate  -f-  and  —  signs  in  the 
column  indicating  elevations.     The  assumed  elevation 
of   the   first  station   is  written   in    the  first  line   of   the 
elevations  column. — see  Table  VI.     The  -j-  S.  on  the  first 
bench   indicates   that   the  line  of   sight   was   1.763   feet 
above  B^\  and  this  added  to  the  elevation  of  B^  gives 
the  elevation  of  the  plane  of  sight,  or,  briefly,  the  height 
of  instrument,  which  is  recorded  in  the  third  column  of 
the  table.     The   readings   on   stations   o,  i,  2,  and  3  in- 
indicate  the  respective  distances  of  these  points  below 
the  line  of  sight;  and  these  quantities  subtracted  from 
the  height  of  instrument  give  their  elevations.     The  in- 
strument is  then  moved,  and  a  sight  is  made  back  to 
station  3.     The^.  S.  on  station  3  added  to  the  elevation 
of  that  station  gives  the  new  height  of  instrument,  from 
which   the  succeeding  minus  sights  are  subtracted  as 
before.     The    table   and    Fig.    77,    page    260,  will    now 
mutually  explain  themselves. 

300.  After  one  page  of  the  book  has  been  filled,  the 
computations    should   be    checked.     When    correct,  the 
difference  of  the  sums  of  the  back-sights  and  fore-sights 
on  the  page  will  be  equal  to  the  difference  between  the 
first  and  last  elevation  on  the  page.     In  this  connection. 


260 


SPIRIT    LEVELS. 


[CHAP,  xi 


the  word  "elevation  "  is  to  be  taken  as  applying  to  the 
elevation  of  a  station  or  the  height 
of  the  instrument.  The  two  eleva- 
tions to  be  used  in  checking  can 
always  be  determined  easily  by  a 
moment's  consideration  of  the  na- 
ture of  the  quantities. 

It  is  to  facilitate  this  method  of 
checking  that  the  minus  sights  on 
turning  points  and  on  intermediates 
should  be  kept  in  separate  columns; 
but  as  a  rule,  in  ordinary  practice, 
all  the  minus  sights  are  recorded  in 
a  single  column.  With  the  excep- 
tion of  the  separation  of  the  minus 
sights,  the  form  shown  in  Table  VI 
is  the  erne  commonly  employed  in 
this  country.  Books  containing  rul- 
ings suitable  for  either  of  these 
forms  are  regularly  sold  by  dealers 
in  engineering  stationery,  the  re- 
marks column  occupying  the  right- 
hand  page,  and  the  remainder  of  the 
table  the  left-hand  page. 

The  computations  of  the  eleva- 
tions of  the  turning  points  and 
bench  marks  should  be  checked  by 
the  rod-man  also,  who  should  keep 
the  notes  of  these  points  for  this 
purpose.  The  elevations  of  the 
turning  points  and  benches  ought 
to  be  computed  and  checked  when 
taken.  The  rod-man  has  ample  time 
to  do  this  while  the  instrument  is 

FIG.   77.— PROFILE    LEVEL- 

ING-  being  moved  forward,  and  the  leveler 

can,   by  watching   his  opportunity,   compute  his  par* 


USING    THE    LEVEL. 


26l 


Remarks. 

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262  SPIRIT    LEVELS,  [CHAP.  Xr 

without  delaying  the  work.     The  elevations  of  the  in- 
termediates may  be  filled  in  afterward. 

Finally,  notice  that  the  above  tests  check  only  the 
computations,  and  in  no  way  prove  the  accuracy  of  the 
field  work. 

301.  Improved  Method  by  Height  of  Instrument.     Some 
engineers    object    to    the    form    of    record    shown    in 
Table  VI,    page   261,  (i)   because  the  station  and    ele- 
vation   columns    are    too   widely   separated   for  conve- 
nience in   platting  the  profile   and  in  referring  to  the 
notes,   and  (2)   because   the  station  column   is  too  far 
from  the  remarks.     For  a  form  which  meets  these  ob- 
jections see  Table  VII,  page  263,  a  form   occasionally 
used  in  railroad  surveying.     The  notes  are  the  same  as 
in   Table    VI, — the  order    of    the    columns    only   being 
changed, — and  therefore  need  no  explanation.     One  of 
the  merits  of  this  form  is  that  wherever  it  is  necessary 
to  combine  two   numbers   by  addition   or   subtraction, 
they  are  found  in  adjacent  columns. 

302.  Form    of  Record  by  Differences.     The  following 
objections  are  sometimes  urged  against  the  method  of 
keeping  level  notes  by  the  height  of  instrument  (§  299- 
301):  (i)  The   elevations  of   the  intermediates   are  not 
checked  at  all  ;  (2)  one  of  the  checks  on  the  benches  and 
turning  points  requires  the  rod-man   to  keep  a  set  of 
notes,  which  he  can  not  well  do  with  a  self-reading  rod; 
and   (3)   the  notes  should    contain  the   data   necessary 
to    check    them    thoroughly  at    any   time.      The    form 
shown  in  Table  VIII,  page  264,  meets  these  objections, 
and  may  be  used  with  either  a  target  or  a  self-reading 
rod,  and  gives  a  check  upon  every  point  and  a  double 
check  upon  turning  points. 

The  first  four  columns  are  copied  from  Table  VI, 
page  261,  and  need  no  further  explanation.  The  quan- 
tities in  the  column  headed  "  Differ."  are  the  differences 
between  the  readings  on  successive  stations.  If  a  read- 


ART.  7] 


USING    THE    LEVEL. 


263 


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264 


SPIRIT    LEVELS. 


[CHAP,  xi 


ing  is  subtracted  from  a  succeeding  one,  the  difference 
is  minus  ;  and  if  from  a  preceding  one,  the  difference  is 

TABLE   VIII. 

FORM  OF  RECORD  FOR  PROFILE  LEVELING. 
Method  by  Differences. 


Sta. 

+  s. 

or 
B.  S. 

-  S. 

Differ. 

Eleva. 

Check. 

Remarks. 

Int's. 

F.  S. 

o 

!.76 

2.52 

—  0.76 

100.00 

99.24 

-9-73 
+  1.76 

£1  highest  point 
of    bowlder,  15 
ft    to  K    of  Sta 

i 

546 

—  2.94 

96.30 

-  7-97 

0  +  10  ft. 

2 

7-83 

-  2.37 

93.93 

IOO.OO 

3 

1-93 

9-73 

-  1.90 

92.03 

92.03 

4 

5.00 

—  3-°7 

88.96 

+  1-93 

5 

7-25 

-  2.25 

86.71 

—  0.79 

6 

8.32 

-1.07 

85.64 

+  1.14 

7 

7-54. 

+  0.78 

86.42 

92.03 

8 

2.17 

0.79 

+  6-75 

93-17 

93-17 

+50 
9 

O.IO 

3-72 

+  2.07 
-3-62 

95-24 
91.62 

-9.2! 
+  2.I7 

Summit  of  Yan- 
kee Ridge. 

10 

II.  OI 

—  7.29 

84-33 

7.04 

2?a 

7-32 

+  3-69 

88.02 

Nail  on  root    of 

II 

3-47 

9.21 

-  1.89 

86.13 

86.13 

tree,  50  ft.  to  R. 

12 

3-40 

IO.2O 

+  0.07 
-  6.80 

86.20 
79.40 

+  3-47 
-0.83 

Bottom  of   Bone 
Yard  Branch,  10 

13 

3.00 

+  7.20 

86.60 

+  2.64 

ft.  wide,   water 

1.72 

+  1.28 

87.88 

86.13 

3  ft.  deep. 

peg 

5.38 

0.83 

+  0.89 

88.77 

88.77 

is 

4.II 

(4-11) 

+  1.27 

90.04 
9.96 

-  2<t    67 

+  14.71 

100.00 

—     9.96 

1 

plus.  The  elevations  are  determined  by  applying  these 
differences,  each  with  its  proper  sign,  to  the  elevation  of 
the  preceding  station. 


ART.    7]  USING    THE    LEVEL.  265 

The  work  is  checked,  as  shown  in  the  column  headed 
"  Check,"  by  taking  the  difference  between  the  B.  S.  and 
F.  S.  for  each  setting  of  the  instrument,  and  applying 
it  to  the  elevation  of  the  preceding  turning  point,  which 
should  give  the  elevation  of  the  succeeding  turning 
point.  Checking  the  elevation  of  the  turning  point 
checks  the  elevation  of  the  preceding  intermediate 
points.  Each  page  may  be  checked  by  adding  the  B.  S. 
and  F.  S.,  and  applying  their  difference  to  the  last  ele- 
vation, in  essentially  the  same  way  as  in  the  other 
forms.  In  Table  VIII  the  elevation  of  station  15  was 
regarded  as  the  "last  elevation,"  and  consequently  the 
reading  on  that  station  was  considered  as  the  last  F.  S., 
which  it  would  be  if  the  work  stopped  at  station  15. 

Notice  that  by  this  method  each  point  is  checked 
once  and  the  turning  points  twice.  Of  course  this 
method  of  keeping  the  notes  requires  more  computing 
than  the  others,  but  it  also  more  thoroughly  checks  the 
work.  If  rapid  work  is  required,  the  quantities  in  the 
"difference"  and  "elevation"  columns  need  not  be 
worked  out  in  the  field,  in  which  case  the  turning  points 
will  be  checked  only  once. 

303.  Drawing  the  Profile.  A  profile  is  a  vertical  sec- 
tion of  the  line  leveled,  showing  the  relative  heights 
and  distances  of  the  various  points  at  which  levels  were 
taken.  It  is  really  a  graphical  representation  of  the 
station  and  elevation  columns  of  the  level  notes,  the  first 
being  the  horizontal  and  the  latter  the  vertical  co-ordi- 
nates. The  ground  may  be  assumed  to  slope  uniformly 
between  the  points  at  which  levels  were  taken,  for  they 
were  taken  not  only  at  the  regular  stations,  but  also  at 
every  considerable  change  of  elevation;  consequently,  a 
line  joining  the  points  obtained  by  plotting  the  level 
notes,  represents  in  detail  the  rise  and  fall  of  the  line, 
as  seen  in  a  side  view. 

The  profile   may  be  made  by  drawing  a  horizontal 


266  SPIRIT    LEVELS.  [CHAP.  XI 

line  to  represent  the  datum,  and  laying  off  along  it,  to 
any  convenient  scale,  the  distances  or  stations  from  the 
first  column  of  the  field  notes.  The  elevations  corre- 
sponding to  these  points  are  next  to  be  laid  off  at  right 
angles  to  the  datum  line  and  above  it.  As  the  rise  and 
fall  of  a  line  is  always  very  small  in  proportion  to  its 
length,  it  is  usual  to  make  the  vertical  scale  much 
greater  than  the  horizontal.  By  thus  employing  two 
different  scales,  the  irregularities  of  the  surface  are 
made  more  apparent  to  the  eye,  and  also  any  subse- 
quent use  of  the  profile  is  rendered  much  easier  and 
more  accurate. 

304.  In  practice,  engraved  profile  paper  is  generally 
used,  which  is  ruled  in  rectangles,  to  which  any  arbi- 
trary values  may  be  assigned.  Three  styles  of  profile 
paper  are  regularly  sold  by  stationers.  They  are  dis- 
tinguished as  A,  B,  and  C,  according  to  the  scale  of  the 
ruling.  The  first  has  four  horizontal  and  twenty  verti- 
cal spaces  to  the  inch  ;  the  second,  four  horizontal  and 
thirty  vertical;  and  the  third,  five  horizontal  and  twenty- 
five  vertical.  The  paper  is  in  rolls  10  yards  long  and 
20  inches  wide,  and  may  be  had  printed  on  either 
opaque  or  translucent  paper. 

It  is  customary  to  make  one  space  horizontally  equal 
to  one  station,  and  one  space  vertically  equal  to  one 
foot.  In  using  engraved  profile  paper  it  is  not  neces- 
sary to  plot  the  datum,  but  only  to  select  a  horizontal 
line  near  the  middle  of  the  paper  and  mark  it,  say,  100 
feet,  and  number  the  other  lines  accordingly.  The 
vertical  lines  being  numbered  to  correspond  with  the 
stations,  the  points  are  easily  and  speedily  plotted.  A 
fine  line  drawn  exactly  through  the  plotted  points  com- 
pletes the  profile. 

If,  owing  to  the  rise  or  fall  of  the  ground,  the  surface 
line  runs  off  the  paper  at  the  top  or  bottom,  it  is  only 
necessary  to  change  the  numbering  of  the  horizontal 


ART.  7]  USING    THE   LEVEL.  267 

lines  and  commence  again  at  the  other  edge.  The  be- 
ginner is  liable  to  plot  the  readings  on  benches,  turning 
points,  and  plus  stations  as  though  they  were  regular 
stations.  The  readings  on  benches  and  turning  points 
are  not  plotted,  except  when  they  are  stations  as  well. 
The  plus  stations  are  plotted  in  their  proper  relative 
position,  the  distance  on  the  profile  being  estimated. 

305.  PRECISE  LEVELING.     Precise,   or  geodesic,   level- 
ing is   differential  leveling  (§  293)   performed  with  the 
utmost  care.     Leveling  of  the  greatest  accuracy  possi- 
ble   is  required  in  surveys  to  determine  the  figure   of 
the  earth,  the  relative  elevations  of  the  surfaces  of  the 
great    lakes,    the    relative    elevation    of    the    water    on 
opposite  sides  of  an  isthmus,  the  fall  of  the  great  rivers, 
etc. 

306.  Methods.     There  are   two   principal  methods  in 
leveling,  according   to  the  sequence  of  the  instrument 
and  rod,  which  may  be  called  single  and  double  leveling. 
Each   may  be   performed   with   one   rod   or  with   two, 
which    gives    rise    to   four   methods,    according   to  the 
sequence   of   the   instrument   and    the  rod.      There  are 
also  two  methods  of  duplicating  the  work:  viz.,  dupli- 
cate leveling  once  over  the  line,  and  duplicate  leveling 
twice  over  the  line.     In  all,  then,  there  are  six  methods 
of  leveling.     The  first  five  are   represented  in   Fig.   78, 
and    the    sixth    is    simply  the    first    method   applied   a 
second  time. 

In  Fig.  78,  7, ,  72,  73,  etc.,  indicate  successive  positions 
of  the  instrument;  Al9Ay,  etc.,  successive  positions  of 
one  rod  ;  B^ ,  7?2 ,  etc.,  successive  positions  of  the  other 
rod;  and  7,  //,  ///,  etc.,  the  several  methods  of  leveling. 

/,  single  leveling  with  one  rod,  is  the  method  employed 
in  ordinary  leveling  (§  293  and  §  296). 

//,  single  leveling  with  two  rods,  is  more  accurate  and 
also  more  rapid  than  leveling  with  a  single  rod.  By 
the  use  of  two  rods  the  back-sight  and  the  fore-sight 


268 


SPIRIT    LEVELS. 


[CHAP,  xi 


may  be  taken  in  quick  succession,  which  saves  time. 
Since  less  time  intervenes  between  the  sights,  there  is 
less  liability  of  change  in  the  plane  of  sight;  and  hence, 
for  this  reason  also,  this  method  is  more  accurate  than 


IT   \ 

A 

I 

/ft 

A. 
i 

I.                     At                    Zt                      A)                     Ij                    A* 

JI       1 

~j( 

/i\ 

M 

A, 

i.               B,               /,               „,                ft           "a 

M  fr  M 

J/T    1 

EZZOUJ 

A 

A. 

t' 

X,                   4.       /)j                  ^                     >«.      ^                    Z)    ^ 

1 

m 

B 

/l\ 

I, 
* 

FIG.  78.— METHODS  OF  LEVELING. 

the  preceding  one.  This  is  the  method  used  on  the 
U.  S.  Lake  Survey  *  and  on  the  Mississippi  River  Sur- 
vey.f 

///,  double  leveling  with  one  rod,  affords  a  perfect  check 
against  errors  of  adjustment  and  observation,  since  the 
difference  of  the  rod  readings  for  the  two  fore-sights 
should  be  the  same  as  the  difference  for  the  two  back- 
sights following. 

IV,  double  leveling  with  two  rods,  combines  all  the  ad- 
vantages of  the  second  and  third  methods.  The  nu- 
merals adjacent  to  the  rods  show  the  order  in  which 


*  Chief  Engineer's  Report,  U.  S.  A.,  for  1880,  pp.  2366 and  2429. 
t  Mississippi  River  Commission  Report  for  1881,  p.  50. 


ART.  7]  USING    THE    LEVEL.  269 

they  are  sighted  upon.  This  is  the  method  employed 
on  the  U.  S.  Coast  and  Geodetic  Survey.*  On  level 
ground  or  where  the  slope  does  not  interfere,  the  dis- 
tances A^A^,  B^A^,  etc.,  are  about  220  meters  (720  feet), 
and  the  distances  B^A^,  A^B^,  etc.,  are  about  20  meters 
(66  feet). 

V,  duplicate  leveling — once  over  the  line, — is  simply  dupli- 
cating the  work  without  the  labor  of  going  twice  over 
the  line.  Two  rods  are  used,  placed  as  in  the  figure. 
After  having  observed  upon  Al  and  B^  the  instrument 
is  pulled  up  and  re-set  a  little  to  one  side  and  the  two 
rods  are  sighted  upon  again.  This  method  duplicates 
the  work  as  far  as  instrumental  errors  are  concerned, 
but  is  not  as  perfect  a  check  as  either  the  third  or  the 
fourth  method. 

A  sixth  method,  duplicate  leveling — twice  over  the  line, — 
consists  in  an  independent  re-leveling  of  the  line.  The 
most  reliable  results  are  obtained  by  repeating  the  work 
in  a  direction  opposite  to  that  in  which  it  was  first  done 

(§315). 

307.  Field  Routine.  After  having  planted  the  tripod 
firmly  and  leveled  the  instrument,  read  both  ends  of 
the  bubble — estimating  the  fraction  of  a  division.  If 
there  is  a  milled-head  screw  under  one  end  of  the  tele- 
scope, the  bubble  can  easily  be  brought  to  the  middle 
each  time  ;  but  if  there  is  not,  it  is  better  to  bring  it 
nearly  to  the  middle  and  apply  a  correction.  It  is  not 
enough  to  read  only  one  end,  since  the  bubble  is  liable 
to  change  its  length  with  a  change  of  position  or  of 
temperature. 

Next  read  the  position  of  the  three  wires  on  the  rod; 
and  then  read  the  bubble  again  for  a  check,  and  also  to 
detect  any  change.  Reverse  the  level  end  for  end,  and 
turn  the  telescope  180°  about  its  optical  axis,  and  repeat 

*  Report  for  1879,  P-  2O^- 


270  SPIRIT    LEVELS.  [cHAP.  XI 

the  operations  as  above.  The  first  reversal  eliminates 
any  inequality  in  the  lengths  of  legs  of  the  striding 
level;  and  the  second  eliminates  any  error  of  collimation. 
The  mean  of  the  several  readings  must  be  corrected  for 
the  difference  in  position  of  the  bubble,  and  for  inequal- 
ity of  the  collars  (§  290). 

Instead  of  reversing  the  level  and  telescope  at  the 
same  time,  the  observations  are  sometimes  made  as  fol- 
lows: read  upon  the  rod,  reverse  the  level,  and  read 
again;  reverse  the  telescope  and  read  a  third  time;  then 
reverse  the  level  and  make  a  fourth  reading.  The  first 
method  is  the  better. 

308.  On  the  Coast  Survey  the  method  of  observing 
differs  slightly  from  that  described  above.*     Errors  of 
level  and  of  collimation  are  eliminated  by  reversing  both 
the  bubble  and   the  telescope  on  each  back-sight  and 
fore-sight.     Each  observation  is  of  a  single  wire  on  a 
target.     The  target  is  set  but  once  for  each  station,  the 
differential   quantities    being  read    by  the  micrometer 
under  the  eye  end  of  the  telescope.     This  seems  not  to 
be  as  good  a  method  as  the  above;  there  are  two  objec- 
tions to  it,  aside  from  the  time  and  labor  required  to 
set    the    target.       First,    there   is    no    sufficient   check 
against  errors  in  reading  the  positions  of  the  target. 
Second,  the  micrometer  is  read  for  a  central  position  of 
the  bubble,  the  telescope  is  then  moved  to  bisect  the 
target,  and  the  screw  is  read  again;  therefore  there  is 
no  check  on  the  stability  of  the  instrument. 

309.  The  Record.     It  is  hardly  wise  to  attempt,  in  this 
volume,  an  explanation  of  the  method  of  recording  the 
notes  and  making  the  computations.     For  a  full  expla- 
nation of  the  method  employed  on  the  U.  S.  Coast  and 
Geodetic  Survey,  see  the  annual  report  of  that  Survey 


*  U.  S.  Coast  and  Geodetic  Survey  Report  for  1879,  PP-  2°6  and  207' 


ART.  7]  USING    THE    LEVEL.  271 


for  1879,  PP-  2I°  an(^  2II>  or  report  for  1880,  p.  139. 
For  a  full  explanation  of  the  form  recommended  by  the 
U.  S.  Lake  Survey,  see  annual  report  of  the  Chief  of 
Engineers,  U.  S.  A.,  for  1880,  Part  III,  p.  2431. 

310.  SOUKCES  OF  ERKOK.     Probably  in  no  other  kind  of 
surveying,  with  the  possible  exception  of  chaining,  is  it 
as  important  to  distinguish  between  compensating  and 
cumulative  errors  (§  18)  as  in  leveling.      In   general  a 
clear   comprehension  of   all  the  sources  of  error,  their 
amounts,  and  the  means  of  avoiding  them,  will  be  of 
great  service  in  indicating  the  care  necessary  to  secure 
a  given  degree  of  accuracy  —  and  particularly  is  this  true 
in  leveling. 

For  convenience  in  discussing  them  we  will  classify 
errors  of  leveling  as  (i)  instrumental  errors,  (2)  rod 
errors,  (3)  errors  of  observation,  (4)  personal  errors, 
(5)  errors  in  recording  and  computing,  and  (6)  errors 
of  curvature  and  refraction.* 

311.  Instrumental  Errors.     The  principal  instrumental 
error  is  that  due  to  the  line  of  sight's  not  being  parallel 
to  the  level.     This  may  be  caused  either  by  imperfect 
adjustment,  or  by  unequal  size  of  rings,  or  by  both.     If 
the  telescope  slide  is  not  straight,  or  does  not  fit  closely, 


*  Owing  to  the  unequal  density  of  the  earth's  crust,  the  plumb-line  does 
not  always  point  to  the  center  of  the  earth;  and  this  gives  rise  to  another 
source  of  error,  i.e.,  the  local  deflection  of  the  plumb-line.  This  source  of 
error  has  been  suggested  as  the  explanation  of  the  discrepancy  in  the  elevation 
above  sea-level  of  a  point  at  St.  Louis,  Mo.,  as  determined  by  two  lines  of 
precise  levels — one  run  from  mean-tide  of  the  Atlantic  Ocean  near  New  York 
City,  and  the  other  from  mean-tide  of  the  Gulf  of  Mexico  near  Mobile,  Ala. 
This  source  of  error  is  too  complicated  for  discussion  here.  For  a  brief  refer- 
ence to  this  subject,  see  U.  S.  Coast  and  Geodetic  Survey  Report  for  1882,  p. 

5i7- 

If  a  line  of  levels  were  run  fron  mean-tide  at  the  equator  to  mean-tide  at 
the  pole,  what  would  the  difference  of  elevation  be?  If  the  difference  of  ele- 
vation were  determined  with  a  mercurial  barometer,  what  would  it  be?  If 
with  an  aneroid  barometer? 


272  SPIRIT    LEVELS.  [CHAP.  XI 

or  does  not  move  in  the  axis  of  the  rings,  it  may  produce 
an  error.  Of  course  the  instrument  must  be  focused  so 
as  to  eliminate  parallax. 

All  of  the  preceding  errors  are  compensating,  and  the 
elevations  of  turning  points  may  be  made  entirely  inde- 
pendent of  them,  whatever  their  value,  by  always  plac- 
ing the  instrument  midway  between  the  turning  points. 

312.  Rod  Errors.  The  principal  rod  error  is  caused  by 
not  holding  the  rod  vertical.  It  is  greater  for  a  large 
rod  reading  than  for  a  small  one.  It  is  compensating 
on  turning  points,  and  may  be  entirely  eliminated  by 
attaching  a  level  or  short  plumb,  or  by  waving  the  rod 
(§  294,  third  paragraph).  In  waving  the  rod  care  must 
be  taken  that  the  face  is  not  lifted  by  the  rod's  resting 
on  its  back  edge  when  revolved  backwards.  To  obviate 
this  source  of  error  and  save  the  time  consumed  in  wav- 
ing the  rod,  a  "  corner  target  "  is  sometimes  used.  This 
consists  of  an  ordinary  target  (Fig.  69  or  Fig.  71, 
page  231),  whose  right  and  left  halves  are  in  planes  at 
right  angles  to  each  other.  The  corner  of  the  rod  and 
of  the  target  face  the  instrument,  and  if  the  rod  is  not 
vertical  the  central  portion  of  the  horizontal  line  of  the 
target  will  be  either  above  or  below  the  horizontal  cross 
hair.  Another  device  for  accomplishing  the  same  pur- 
pose is  a  target  similar  to  Fig.  69  or  Fig.  71,  in  which 
the  right  and  left  thirds  are,  say,  2  inches  behind  the 
plane  of  the  middle  third.  If  the  rod  is  not  vertical, 
the  horizontal  line  on  the  middle  third  of  the  target  will 
be  above  or  below  that  on  the  side  portions.  For 
several  reasons  neither  of  these  devices  is  as  rapid  or  as 
accurate  as  a  short  plumb-line  or  level — either  a  disk 
level,  or  two  level  vials  at  right  angles  to  each  other 
— attached  to  the  rod. 

With  telescoping  target  rods,  when  extended,  the 
slipping  of  the  upper  piece  after  the  target  has  been 
pronounced  correct  and  before  the  vernier  has  been 


ART.   7]  USING    THE    LEVEL.  273 

read,  is  a  source  of  error.  The  target  itself  may  slip, 
but  this  is  not  so  probable,  because  of  its  inferior  weight. 

Another  source  of  error  is  the  settling  of  the  turning 
point,  due,  in  coarse  or  sandy  soil,  to  its  own  weight,  or 
to  the  impact  of  setting  the  rod  upon  it.  The  resulting 
error  is  cumulative.  The  remedy  in  the  first  case  is  to 
use  a  long  peg,  or  to  rest  the  rod  upon  a  triangular 
plate  having  the  corners  turned  down  slightly  (§  294, 
fifth  paragraph).  Whatever  the  turning  point,  the  rod 
should  never  be  dropped  upon  it. 

Finally,  another  small  rod  error  is  the  error  in  the 
graduated  length.  This  affects  only  the  total  difference 
of  elevation  between  the  two  points.  With  the  numer- 
ous home-made  self-reading  rods  now  in  use,  this  is  a 
much  more  important  source  of  error  than  with  the  rods 
made  by  professional  instrument  makers. 

"  An  important  source  of  error  in  spirit  leveling,  and 
one  very  commonly  overlooked,  is  the  change  in  the 
length  of  the  leveling  rod  from  variations  of  tempera- 
ture. It  is  quite  possible  that  errors  from  this  source 
may  largely  exceed  the  errors  arising  from  the  leveling 
itself."* 

313.  Errors  of  Observations.  The  principal  error  of  ob- 
servation is  in  reading  the  position  of  the  bubble.  Even 
if  the  bubble  is  kept  in  the  middle,  it  is  nevertheless 
read.  Every  leveler  should  know  the  error  on  the  rod 
corresponding  to  a  given  difference  of  reading  of  the 
bubble,  since  he  then  knows  how  accurately  he  must 
read  the  bubble  for  the  particular  accuracy  aimed  at. 

A  small  error  arises  from  the  fact  that  the  adhesion 
of  the  liquid  to  the  sides  of  the  glass  tube  prevents  the 
bubble  from  coming  precisely  to  its  true  point  of  equi- 
librium (see  the  third  paragraph  of  §  257).  Even  though 
it  may  finally  arrive  at  the  true  point,  it  is  liable  to  be 

*  Wright's  Adjustments  of  Observations,  p.  372. 


274  SPIRIT    LEVELS.  [CHAT.  XI 

read  before  it  has  stopped  moving.  The  bubble  should 
be  re-read  after  the  target  is  nearly  adjusted,  or,  with  a 
self-reading  rod,  after  the  reading  has  been  taken  and 
before  the  rod-man  is  signaled  to  move  on. 

Another  small  error  is  due  to  the  effect  of  the  sun  in 
raising  one  end  of  the  telescope  by  the  unequal  expan- 
sion of  the^different  parts  of  the  instrument.  In  ordi- 
nary leveling  operations,  the  bubble  is  first  brought  to 
the  middle  and  then  the  target  is  sighted  in,  leaving  an 
interval  for  the  sun  to  act.  The  error  is  greatest  in 
working  toward  or  from  the  sun.  Ordinarily  it  is  cu- 
mulative; for  on  the  back-sight  one  wye  is  expanded, 
which  elevates  the  line  of  sight,  while  on  the  fore-sight 
the  other  wye  is  expanded,  which  depresses  the  line  of 
sight,  the  two  errors  affecting  the  difference  of  elevation 
in  the  same  way.  The  error  on  the  fore-sight  is  farther 
increased  by  the  cooling  of  the  wye  which  was  expanded 
during  the  back-sight.  The  error  due  to  the  sun  is 
always  small,  and  can  be  nearly  eliminated  by  noticing 
the  position  of  the  bubble  after  setting  the  target;  and 
can  be  still  farther  reduced  by  shading  the  instrument, 
although  "  the  Indian  Geodetic  Survey  proved  conclu- 
sively that  the  error  was  appreciable,  even  when  the 
instrument  was  shaded." 

Such  a  seemingly  trivial  thing  as  the  unequal  heating 
of  the  bubble  tube  produces  an  appreciable  effect.  The 
bubble  always  moves  toward  the  warmer  portion  of  the 
tube,  owing  to  a  change  in  adhesion  of  the  fluid  to  the 
glass  tube;  and  therefore,  in  accurate  work,  the  leveling 
instrument  should  be  protected  from  the  direct  rays  of 
the  sun,  and  the  bubble  tube  should  never  be  touched 
with  the  finger  nor  be  breathed  upon. 

If  the  observer  is  compelled  to  change  his  position 
after  reading  the  level  and  before  sighting  the  telescope, 
his  movement  about  the  instrument  may  cause  a  change 
in  the  inclination  of  the  telescope.  "  In  some  trials  in 


ART.  7]  USING    THE    LEVEL.  275 

France,  the  inclination  from  this  cause  (one  leg  being 
in  the  line  of  sight)  varied  from  two  to  one  hundred 
seconds  of  arc  according  as  one  leg  rested  on  the  pave- 
ment or  on  vegetable  soil."  *  This  error  is  a  minimum 
when  two  legs  are  placed  in  a  line  parallel  to  the 
route  and  the  third  at  right  angles  to  it.  Sometimes,  to 
eliminate  this  error,  one  man  reads  the  bubble  while  a 
second  sights  the  telescope;  but  this  is  objectionable, 
since  it  requires  two  skillful  men  and  also  divides  the 
responsibility.  It  is  better  to  provide  the  instrument 
with  a  mirror  or  a  prism  so  that  the  bubble  may 
be  read  from  the  position  from  which  the  telescope 
is  sighted;  for  when  the  bubble  is  almost  constantly  in 
motion,  the  same  man  should  see  it  and  the  cross  hairs 
at  the  same  time. 

It  has  been  found  that  appreciable  errors  are  caused 
by  the  settling  of  the  instrument  on  its  vertical  axis, 
and  by  the  settling  of  the  tripod  legs  into  the  ground. 
In  spongy  or  clayey  soil  the  tripod  legs  are  some- 
times gradually  lifted  up.  These  errors,  though  small 
in  themselves,  are  more  important  than  is  generally 
supposed,  inasmuch  as  they  are  cumulative  (§  18).  They 
can  be  eliminated  by  re-running  the  line  in  the  opposite 
direction. 

If  the  reading  is  made  on  either  side  of  the  vertical 
hair,  there  is  a  possibility  of  error,  owing  to  the  horizon- 
tal hair's  not  being  horizontal;  and  with  wye  levels  not 
provided  with  a  means  of  preventing  the  rotation  of  the 
telescope  in  the  wyes,  this  possibility  becomes  a  probabil- 
ity. Any  device  which  insures  the  horizontality  of  the 
horizontal  hair  increases  the  rapidity  and  accuracy  of 
the  work.  This  error  is  compensating. 

The  inaccuracy  of  telling  when  the  hair  exactly 
covers  the  center  of  the  target  is  a  source  of  slight  error, 

*  Wilfred  Airy,  in  Proc.  Inst.  of  C.E.,  Vol.  78,  p.  458. 


276  SPIRIT    LEVELS.  [CHAP.  XI 

but  not  so  slight  as  many  think.  In  setting  a  quadrant 
level  target,  the  instrument  having  a  telescope  magni- 
fying thirty-five  times  and  a  bubble  tube  with  a  radius 
of  145  feet,  the  average  probable  error  (§  2  of  Appendix 
III)  for  a  class  of  fifteen  was  1.4  thousandths  of  a  foot 
at  100  feet  and  2.25  thousandths  of  a  foot  at  300  feet 
(see  Tables  I  and  II,  Appendix  III).  In  running  a  line 
of  levels  this  error  would  be  compensating.  Owing  to 
this  source  of  error,  the  difference  in  accuracy  between 
a  target  rod  and  a  self-reading  rod  is  not  so  great  as 
their  difference  in  precision. 

With  a  target  rod,  errors  of  one  foot,  one  tenth,  etc., 
are  not  uncommon.  In  double  and  duplicate  leveling 
(§  3°6)  these  errors  are  easily  discovered  and  corrected. 
In  ordinary  leveling  (§§  293  and  296)  they  may  be  elimi- 
nated by  having  the  rod-man  and  instrument-man  read 
the  rod  independently  and  afterwards  compare  notes. 
In  single  differential  leveling  with  one  rod  both  men 
can  read  the  rod  without  materially  delaying  the  work, 
by  the  rod-man's  reading  the  rod  and  recording  it  in  a 
book  carried  for  that  purpose,  and  carrying  the  rod  to 
the  leveler  if  it  be  a  back-sight  or  waiting  for  the  com- 
ing of  the  leveler  if  it  be  a  fore-sight,  when  the  observer 
also  reads  the  rod  and  records  it  in  his  book,  after  which 
the  two  records  are  compared.  But  in  profile  leveling 
this  can  not  be  done;  and  the  only  way  to  check  the 
work  is  to  duplicate  it.  With  a  self-reading  rod  the 
liability  of  this  class  of  errors  is  somewhat  reduced  by 
reading  three  hairs. 

314.  Personal  Errors.  The  errors  previously  described 
are  liable  to  occur  with  any  observer.  They  are  due 
chiefly  to  the  instruments  and  to  the  nature  of  the  work, 
and  would  probably  not  materially  differ  for  equally 
skilled  observers.  We  come  now  to  a  class  of  errors 
which  depend  mainly  upon  inaccuracies  peculiar  to  the 
individual.  One  observer  may  read  a  target  higher  or 


ART.   7]  USING     THE    LEVEL.  277 

lower  than  another  of  equal  skill;  or  in  reading  the  posi- 
tion of  the  bubble  he  may  have  peculiar  views  as  to 
what  constitutes  the  end  of  the  bubble;  or  he  may  ha- 
bitually read  the  bubble  so  as  to  get  a  distorted  view  of 
it  through  the  glass  tube.  With  skillful  observers,  all 
such  errors  are  quite  small  and  generally  cancel  them- 
selves. In  fact,  the  errors  here  classed  as  personal  are 
possible  rather  than  demonstrated  as  actually  occurring; 
and  yet  there  is  nothing  more  certain  than  that  in  any 
series  of  accurate  observations  there  is  a  difference  be- 
tween the  results  obtained  by  different  individuals. 
This  difference  is  known  as  the  personal  equation.  In 
long  lines  of  accurate  leveling,  it  has  been  found  that 
each  man's  way  of  performing  the  several  operations  has 
an  appreciable  effect  upon  the  final  result. 

315.  It  is  a  curious  fact,  but  one  abundantly  verified, 
that  when  lines  are  duplicated  in  opposite  directions, 
the  discrepancies  tend  to  one  sign  and  increase  with  the 
distance.  This  subject  has  been  much  discussed,  and 
various  explanations  of  the  fact  have  been  offered;  as 
settling  of  instrument,  settling  of  turning  point,  dis- 
leveling  effect  of  the  sun,  unequal  illumination  of  the 
target  on  back-sights  and  fore-sights,  unequal  illumina- 
tion of  the  two  ends  of  the  bubble,  effect  of  refraction 
in  reading  the  bubble,  the  change  of  the  position  of  the 
observer  to  read  the  bubble  (see  §  313,  fifth  paragraph), 
and  personal  bias  in  reading  the  target  or  bubble;  but 
none  of  these  reasons  are  entirely  satisfactory.  These 
discrepancies  vary  with  different  observers;  are  not  even 
constant  for  the  same  observer;  are  nearly  proportional 
to  the  distance;  and  seem  to  be  independent  of  the 
nature  of  the  ground,  the  direction  in  which  the  work  is 
done,  the  season,  or  the  manner  of  supporting  the  rod.* 

*  For  a  valuable  discussion  of  this  source  of  error,  together  with  numerical 
results  derived  from  actual  work,  see  Report  of  the  Mississippi  River  Commis- 
sion for  1883,  pp.  141-62  ;  or  Report  of  Chief  of  Engineers  U.  S.  A.,  for  1884, 
PP-  2547-68, 


278  SPIRIT    LEVELS.  [CHAP.  XI 

The  effect  of  this  class  of  errors  maybe  eliminated  by 
each  observer's  duplicating  his  work  in  the  opposite 
direction  under  as  nearly  the  same  conditions  as  pos- 
sible. The  accuracy  is  increased  by  leveling  alternate 
sections  in  opposite  directions,  as  is  done  in  India; 
and  it  may  be  still  further  increased  by  reading  the 
back-sight  first  each  alternate  time  the  instrument  is 
set  up. 

316.  Errors  of  Recording  and  Computing.     Such  blun- 
ders as  recording  the  fore-sight  in  the  back-sight  column 
and,  vice  versa,  the  back-sight  in  the  fore-sight  column, 
although  the  result  of  gross  carelessness,  do  neverthe- 
less occur.     To  check  against  such  errors,  the  rod-man 
and  the  leveler  should  read  the  rod  independently,  as 
explained   in   the   last  paragraph  of    §  313.     In   profile 
leveling,  errors  of  this  class  are  easily  discovered,  if  the 
rod-man  keeps  the  notes  and  works  out  the  elevations 
of  turning  points  and  benches;  and  in  any  case,  such 
errors  are  more  likely  to  be  discovered  if  the  elevations 
are  worked  out  immediately  after  the  observations  are 
taken. 

Errors  of  computation,  and  the  method  of  checking 
them,  have  already  been  discussed,  incidentally,  in 
§§  299-302,  which  see. 

317.  Errors  of  Curvature  and  Refraction.     The  object 
of  leveling  is  to  find  the  distance  that  one  point  is  above 
or  below  a  level  surface  (§  292)  passing  through  some 
other  point.     The  line  of  sight  of  a  properly  adjusted 
leveling  instrument  is  a  horizontal  line,  i.e.,  a  tangent 
to  a  level  line.     Therefore,  if  the  two  points  sighted  at 
are  not  equidistant  from  the  instrument,  the  difference 
between  the  rod  readings  will  not  be  the  true  difference 
of  level,  owing  to  the  curvature  of  a  level  surface,  i.e., 
to  "  the  curvature  of  the  earth."     The  difference  between 
a  level  and  a  horizontal  line  can  be  computed  (§  319); 
and  hence  if  the  length  of  sight  is  known,  the  effect  of 


ART.  7]  USING    THE    LEVEL. 


curvature  can  be  eliminated  by  applying  a  correction. 
The  error  is  compensating,  and  may  be  entirely  elimi- 
nated by  setting  the  instrument  midway  between  turn- 
ing points. 

318.  Owing  to  the  refraction  of  the  atmosphere  the 
beam  of  light  from  the  target  to  the  telescope  is  slightly 
concave  downwards,  and  hence  the  line  of  sight  is  not 
a  truly  horizontal  line.  The  difference  between  a  hori- 
zontal line  and  the  true  line  of  sight  can  be  computed 
for  an  average  condition  of  the  atmosphere  (§  319)  ;  and 
therefore  if  the  length  of  sight  is  known,  and  if  the  air 
is  in  its  normal  condition,  the  effect  of  refraction  can 
be  eliminated  by  applying  a  correction. 

But  the  atmosphere  is  not  always  in  its  normal  con- 
dition; and  hence  if  there  is  abnormal  refraction  or  a 
change  of  refraction  between  sights,  there  may  be  re- 
sidual errors  of  refraction,  even  though  the  correction 
for  mean  refraction  be  applied.  As  refraction  varies 
greatly  with  the  nature  of  the  surface  over  which  the 
light  passes  and  also  with  the  distance  from  the  surface, 
the  effect  of  the  refraction  may  vary  considerably.  The 
most  serious  effect  of  variable  refraction  is  the  tremu- 
lousness  or  "boiling"  of  the  atmosphere  caused  by  the 
innumerable  small  currents  of  air  of  different  tempera- 
tures or  densities  which  exist  when  the  atmosphere  and 
the  earth  differ  much  in  temperature.  The  beam  of 
light  in  proceeding  from  the  target  to  the  telescope 
passes  alternately  through  denser  and  rarer  media,  each 
of  which  produces  a  slight  refraction  of  the  ray,  thus 
causing  the  target  to  "  dance  "  and  making  it  difficult 
to  determine  just  when  it  is  properly  bisected  by  the 
cross  hair.  When  this  condition  exists  the  only  remedy 
is  to  shorten  the  length  of  sight,  or  wait  for  better 
atmospheric  conditions.  The  atmosphere  is  usually  in 
the  best  condition  for  seeing  just  before  sunrise  and  a 


SPIRIT    LEVELS. 


[CHAP,  xi 


little  while  before  sunset,  although  the  refraction  is  then 
greater.  A  cloudy  day  is  better  than  a  clear  one. 

Ordinarily  this  error  is  compensating;  and  will  usu- 
ally be  eliminated  by  setting  the  instrument  midway 
between  the  turning  points. 

319.  Correction  for  Curvature  and  Refraction.  To 
compute  the  correction  for  curvature  of  the  earth,  let 
AD,  Fig.  79,  represents  the  horizontal  line  ;  AB  the 
level  line  ;  and  AE  the  radius  of  the  earth.  Then  DB 
is  the  correction  for  curvature.  By  geometry  AD*  = 
DB(2AE  -f  DB].  Dropping  DB  from  the  parenthesis, 
as  it  is  very  small  in  comparison  with  zAE,  and  repre- 
senting the  length  of  sight  by  k  and  the  radius  of  curva- 

k* 
ture  of  the  earth  by  p,  BD  =  —  .     If  BD  is  expressed  in 

feet  and  k  in  miles,  the  correction  for  curvature  becomes 
BDit.  =  0.66  7/fca  miles  .....     (i) 

To  compute  the  correction  for  refraction,  notice  that 
D'  ,  Fig.  79,  is  the  true  position 
of  the  target  and  D  its  apparent 
position.  Hence  DD'  is  the 
correction  sought.  The  ratio  of 
the  refraction  angle,  DAD'  ,  to 
the  angle  at  the  center  of  the 
earth,  AED,  is  known  as  the 
coefficient  of  refraction,  which 
we  will  represent  by  m. 


k 

AED,  in  sec.  of  arc,  =  —  :  ---  r.  . 
psm  i" 

DD'  = 


FIG.  79. 


=  kD'AD"\.zm"  = 


The  ordinary  values  of  m  vary  between  0.06  and  0.08, 


ART.  7]  USING  THE  LEVEL.  281 

The  value  generally  employed  in  computing  the  correc- 
tion for  refraction  is  0.07.     Then 

D'D  ft.  —  0.09/P  miles  .....     (2) 
Combining  equations  (i)  and  (2),  we  have 

BD'  ft.  =  0.57/fc2  miles  .....     (3) 


In  general,  the  total  correction  for  curvature  and  re- 
fraction, to  be  applied  to  the  observed  reading,  is 

ft* 
BD'  =  BD  —  DD'  =  (i  -  2m)—  =  0.000,000,020,45^. 

If  k  —  220  ft,  BD'  —  o.ooi  ft.;  for  k  =  300  ft.,  BD'  = 
0.002  ft.  Notice  that  the  correction  varies  as  the  square 
of  the  length  of  sight.  Numerous  tables  have  been 
computed  which  give  this  correction  directly  for  the 
different  lengths  of  sight  ;  or  tables  can  be  prepared 
which  will  give  the  difference  of  the  correction  with  the 
difference  of  length  of  sight  for  an  argument. 

320.  LIMITS  OF  PRECISION.  The  probable  error  per 
unit  of  distance  is  generally  adopted  as  a  convenient 
measure  of  the  precision  reached.  According  to  the 
theory  of  probabilities,  the  final  error  of  a  series  of  obser- 
vations, affected  only  by  accidental  errors,  will  vary  as 
the  square  root  of  the  number  of  observations.  In 
leveling,  a  method  should  be  adopted  which  will  elimi- 
nate all  cumulative  errors  ;  and  therefore,  since  only 
compensating  errors  remain,  the  final  error  of  leveling 
a  number  of  units  of  distance  is  assumed  to  vary  as  the 
square  root  of  the  distance. 

This  assumption  would  be  true  if  only  compensating 
errors  remained  uncorrected,  and  if  the  number  of  ob- 
servations were  strictly  proportional  to  the  distance 
leveled,  i.e.,  if  the  length  of  sight  was  constant  and  if 


282  SPIRIT    LEVELS.  [CHAP.  XI 

the  inclination  of  the  surface  of  the  ground  leveled  over 
was  the  same  (see  Table  IX,  page  283).  Since  it  is 
improbable  that  all  cumulative  errors  will  be  entirely 
eliminated,  that  part  of  the  final  error  which  is  due  to 
cumulative  errors  will  vary  as  the  distance  leveled.  It 
has  frequently  been  noticed  that,  considered  individu- 
ally, the  errors  of  a  number  of  short  lines  were  well 
within  the  limits  which  were  prescribed  to  vary  as  the 
square  of  the  distance  ;  yet  when  the  sum  for  several 
lines  were  considered,  the  total  discrepancy  would  exceed 
the  limit.  In  other  words,  the  error  is  not  strictly  pro- 
portional to  the  square  root  of  the  distance.  One  part 
of  the  error  is  proportional  to  the  square  root  of  the 
distance,  and  another  portion  varies  nearly  as  the  dis- 
tance; hence,  the  shorter  the  distance,  the  easier  to  attain  a 
limit  prescribed  to  vary  as  the  square  root  of  the  distance. 

If  the  error  was  determined  by  duplicating  the  work 
in  the  same  direction,  and  especially  if  at  the  same  time, 
by  methods  III,  IV,  or  V,  Fig.  78,  page  268,  the  differ- 
ence will  be  the  apparent  error,  and  necessarily  be  too 
small.  The  result  obtained  by  the  adjustment  of  a  net 
of  lines  by  the  method  of  least  squares  affords  the  best 
means  of  determining  the  degree  of  precision. 

321.  According  to  the  Geodetic  Association  of  Europe, 
levels  of  precision  executed  of  late  years  in  Europe  show 
that  the  probable  error  of  a  line  of  levels  of  precision 
should  never  exceed  5  mm.  ^distance  in  kilometers 
(0.0208  ft.  Smiles);  that  3  mm.  4/dist.  in  kilometers  is 
tolerable,  2  mm.  Vdist.  in  kilometers  is  a  fair  average, 
and  i  mm.  Vclist.  in  kilometers  is  high  precision.*  The 
Coast  Survey  requires  5  mm.  4/2  dist.  in  kilometers 
(0.030  ft.  Vmiles).  The  Mississippi  River  Commission's 
limit  is  5  mm.  4/dist.  in  kilometers  (0.021  ft.  Vmiles). 

Of    late   years    the    Coast    Survey's   and    Mississippi 

*  Report  of  U.  S.  Coast  and  Geodetic  Survey  for  1882,  page  522. 


ART.  7] 


USING    THE    LEV2L. 


283 


River  Survey's  work  are  considerably  within  the  limit 
of  2  mm.  Vdist.  in  kilometers.  Table  IX  shows  the 
degree  of  accuracy  attained  in  precise  leveling  on 
the  national  surveys  of  Great  Britain,  India,  and  Switz- 
erland.* 

TABLE  IX. 

DATA  SHOWING  THE  DEGREE  OF  ACCURACY  ATTAINED  IN  PRECISE 
LEVELING,  AND  THE  EFFECT  OF  DIFFERENT  INCLINATIONS  OF 
JHE  GROUND. 


Reference  No. 

KIND  OF  GROUND. 

AVERAGE  DIFFERENCE  PER 
MILE  BETWEEN  Two 
OBSERVERS.               . 

Great 
Britain. 

India. 

Switzer- 
land. 

I 
2 

3 
4 
5 

Nearly    level,    with     very     favorable 
weather                                         . 

foot. 
0.0230 
0.0238 
0.0379 
0.0566 

foot. 
0.0142 
0.0168 
0.0208 
0.0350 

foot. 
0.0125 
0.0148 
0.0183 
0.0308 
0.0416 

Slightly  undulating  —  gradients  not  ex- 

Gradients  entirely  between   I  in   100 
and  i  in  20                                        . 

Gradient  entirely  between  I  in  20  and 
i  in  10     

Steep   and    rough    ground  —  gradients 
frequently  steeper  than  i  in  10  .     . 

To  attain  the  preceding  limits  requires  skillful  ob- 
servers, the  best  instruments,  and  plenty  of  time.  Ordi- 
narily there  are  not  more  than  three  or  four  hours  of  the 
day  in  which  this  class  of  work  can  be  done;  and  as  a 
general  average  not  more  than  one  or  two  miles  can  be 
run  in  a  day  (see  §  323). 

322.  It  is  impossible  to  establish  a  limit  for  work  less 
accurate  than  the  best,  since  the  conditions  under 
which  it  may  be  done  are  too  diverse.  However,  the 
difference  in  precision  between  ordinary  leveling  and 
precise  leveling,  is  not  as  great  proportionally  as  the 
difference  in  care  and  time  given.  A  little  increase  in 

*  Wilfred  Airy  in  Proc.  Inst.  of  C.  E.,  Vol.  44,  page  181. 


SPIRIT    LEVELS.  [CHAP.  XI 


accuracy  costs  a  very  great  increase  of  effort.  Results 
of  leveling  of  apparently  greater  accuracy  than  the  fore- 
going are  often  given,*  but  an  occasional  accurate  result 
—  probably  more  largely  due  to  good  fortune  than  good 
management  —  gives  no  indication  as  to  what  results 
may  be  regularly  expected. 

Regularity  of  result  and  evenness  of  error  is  of  more 
importance  than  occasionally  a  small  disagreement. 
Naturally,  it  is  usually  the  latter  that  is  reported. 

A  line  of  "ordinary"  levels  was  run  on  the  bank  of 
the  Mississippi  River,  and  checked  upon  the  benches  of 
the  precise  levels,  with  an  average  error  of  0.034  feet 
Vdist.  in  miles.  f  The  conditions  under  which  this  work 
was  done  were  about  the  same  as  those  of  a  prelimi- 
nary railroad  survey,  but  it  is  probably  more  accurate 
than  such  surveys  usually  are. 

In  some  of  the  branches  of  the  A.,  T.,  &  S.  F.  R.  R., 
the  instructions  were  to  re-run  the  line  whenever  the 
difference  between  the  levels  on  construction  and  loca- 
tion was  0.03  foot,  between  benches  about  2,000  feet 
apart.  This  is  equivalent  to  limiting  the  maximum 
admissible  error  to  0.048  feet  tMist.  in  miles. 

In  the  topographical  survey  of  the  city  of  St.  Louis, 
Mo.,  "  the  limit  of  error  allowed  was  5  millimeters 
V'dist.  in  kilometers  (0.0208  feet  4/dist.  in  miles).  The 
average  closure  was  0.013  ^eet  ^dist.  in  miles.  The 
probable  error  in  the  determination  of  a  single  mile  of 
the  work  was  o.ooi  foot."  J 

323.  Speed.     The  amount  of  work  that  an  observer 


*  For  example,  several  text-books  contain  the  statement:  "A  French 
leveler  contracts  to  level  the  bench  marks  of  a  railroad  survey  to  within  0.002 
ef  a  foot  per  mile."  Compare  this  with  §  321  and  also  with  Tables  I  and 
II  of  Appendix  III. 

t  Report  of  Mississippi  River  Commission  for  1882,  p.  2269 ;  or  Report  of 
Chief  of  Engineers,  U.  S.  A.,  for  1883,  p.  2269. 

J  The  Technograph,  No.  5  (1890-91),  p.  n. 


ART.   7]  USING    THE    LEVEL.  285 

should  do  in  a  day  can  not  be  stated  definitely.  It 
depends  upon  the  accuracy  required,  the  power  and 
delicacy  of  the  instrument,  the  method  pursued,  the 
ground,  and  very  largely  upon  the  atmospheric  condi- 
tions. For  the  very  best  work,  not  more  than  three  or 
four  hours  can  be  utilized,  even  in  clear  weather.  The 
average  daily  run  for  several  seasons  on  the  Mississippi 
River,  using  a  Kern  level  and  method  II  (§  306),  was 
one  and  a  half  miles  a  day  for  the  entire  season,  and 
two  and  a  half  miles  for  the  days  on  which  work  was 
actually  done.*  On  the  U.  S.  Lake  Survey,  with  the 
same  instrument  and  method,  the  distance  was  about 
two  miles  a  day  for  the  days  on  which  leveling  was 
done.f  On  the  Swiss  levels  of  precision,  3  kilometers 
(1.8  miles)  along  railroads,  and  2  kilometers  (1.2  miles) 
along  highways  in  the  plains,  was  considered  a  fair 
day's  work.J 

Professor  J.  B.  Johnson, §  who  has  had  large  experi- 
ence in  levels  of  precision  on  the  U.  S.  Lake  Survey 
and  on  the  Mississippi  River,  states  that  "  with  a  wye 
level  and  a  target  rod,  a  single  instrument  should 
duplicate  thirty  miles  per  month,  with  no  greater  error 
than  0.05  feet  Vdist.  in  miles,  or  with  a  level  of  precision 
and  speaking  rod,  do  the  same  work  with  a  limit  of  0.02 
feet  i^dist.  in  miles." 

324.  PRACTICAL  HINTS.  For  a  number  of  points  ap- 
plicable in  leveling,  particularly  to  setting  the  tripod, 
see  §  128. 

If  the  instrument  has  once  been  leveled,  and  the 
bubble  is  found  to  have  moved  a  little,  bring  it  back 
with  a  slight  pressure  of  the  finger. 


*  Report  of  Chief  of  Engineers,  U.  S.  A.,  for  1884,  p.  2462. 
t  Professional    Papers   Corps   of   Engineers,  U.  S.  A.,    No.    24 — Primary 
Triangulation  U.  S.  Lake  Survey, — pp.  597-99. 
%  Proc.  Inst.  of  C.  E.,  Vol.  78,  p.  456. 
§  Jour.  Association  of  Engineering  Societies,  Vol.  2,  p.  160. 


286  SPIRIT    LEVELS.  [CHAP.  XI 

Instruments  are  usually  provided  with  sun-shades  to 
prevent  trouble  from  the  sun's  shining  into  the  tele- 
scope. If  the  metallic  shade  is  not  at  hand,  make  one 
by  rolling  up  a  piece  of  paper  and  gumming,  pinning  or 
tying  it  together,  or  springing  a  rubber  band  around  it. 
This  is  easier  and  better  than  holding  the  hat  or  note- 
book over  the  objective. 

In  ascending  or  descending  a  steep  hill,  it  is  desir- 
able, for  speed,  that  the  line  of  sight  should  strike  as 
near  as  possible  to  the  bottom  of  the  rod  on  the  up-hill 
side,  and  to  the  top  ©f  the  rod  on  the  down-hill  side. 
In  selecting  the  position  of  the  instrument  to  satisfy 
this  condition,  set  the  instrument  up  lightly,  turn 
the  telescope  in  the  right  direction,  bring  the  bubble 
approximately  to  the  middle  by  manipulating  the 
tripod  legs,  and  .sight  along  the  outside  of  the  tele- 
scope. Even  this  rude  observation  will  be  valuable  as 
showing  whether  the  instrument  should  be  moved  up 
or  down  the  hill,  and  it  will  save  considerable  time. 
With  a  little  practice,  the  same  observation  may  be 
made  by  drawing  the  tripod  legs  together  and  using 
them  as  a  Jacob's  staff,  when  the  bubble  can  speedily  be 
brought  to  its  proper  position  by  simply  inclining  the 
whole  instrument. 

If  the  up-hill  rod  is  too  near  to  be  focused  on,  move 
either  the  rod  or  the  instrument  a  little  to  one  side.  In 
short  intermediate  sights  for  which  the  telescope  can  not 
be  focused,  it  is  sufficient  to  sight  by  the  bottom  of  the 
wyes  or  by  the  side  of  the  telescope. 

In  closing  at  noon  or  night,  be  careful  to  set  half-way 
between  the  last  two  turning  points;  and  on  resuming 
work,  set  near  one  of  these  points,  and  re-determine 
their  difference  of  level.  The  same  difference  of  level 
each  time  affords  an  excellent  check  upon  the  adjust- 
ments of  the  instrument  (§  234). 


ART.  7]  USING    THE    LEVEL.  287 

325.  Length  of  Sight.     The  length  of  sight  is  limited 
by  the  power  of  the  telescope,  the  atmospheric  condi- 
tions,   the    accuracy    desired,    the    time    available,    etc. 
Some  errors  increase  directly,  and  others  indirectly,  as 
the  number  of  sights  taken   in  a  given   distance.     It  is 
generally  assumed  that  for  the    most  accurate  work  the 
rod  should  be  at  least  100  feet  from  the  instrument  and 
never  more   than  400  feet;  and    that  for  ordinary  work 
the  rod  should  be  300  or  400  feet  away  and  never  more 
than  500  or  600  feet. 

On  the  U.  S.  Coast  and  Geodetic  Survey  the  length 
of  sight  ranges  from  50  to  150  meters  (164  to  492  feet) 
according  to  the  condition  of  ground  and  weather,  the 
average  being  no  meters  (360  feet),  the  distance  be- 
tween the  two  rods  on  the  same  side  of  the  instrument 
being  20  meters  (see  method  IV,  Fig.  78,  p.  268).  On 
the  U.  S.  Lake  Survey*  the  maximum  length  of  sight 
was  100  meters  (328  feet).  On  the  Prussian  Land  Survey, 
the  maximum  sight  has  not  exceeded  50  meters  (164 
feet),  except  for  river  crossings.!  In  the  Swiss  levels 
of  precision,  the  length  of  sight  on  railroads  with  gradi- 
ents under  i  in  100  was  100  meters  (328  feet);  on  rail- 
roads with  steeper  gradients,  from  50  to  100  meters  (165 
to  328  feet);  on  highways  in  the  plains,  from  30  to  60 
meters  (100  to  200  feet);  and  on  mountain  roads,  from 
10  to  25  meters  (33  to  82  feet).J 

326.  Equal  Back-sight  and  Fore-sight.     It  is  very  desir- 
able that  at  each  setting  of  the  instrument  the  lengths 
of  the  back-sight  and  the  fore-sight  should  be  equal  ; 
for,  as  has  been  seen,  there  are  then  a  number  of  errors 
which  cancel  each  other.      This  is  a  very  important  point, 

*  Professional  Papers  Corps  of  Engineers,  U.  S.  A.,  No.  24 — Primarjr 
Triangulation  U.  S.  Lake  Survey, — p.  598. 

t  Wright's  Adjustment  of  Observations,  p.  375. 
^  Proc,  Jnst.  of  C,  E,,  Vol.  78,  p.  456, 


288  SPIRIT    LEVELS.  [CHAP.  XI 

and  should  always  be  kept  in  mind.  Leveling  is  the  only 
kind  of  surveying  wherein  the  instrumental  errors  may 
be  thoroughly  eliminated  without  duplicating  the  work. 
This  is  done  by  making  the  back-sights  and  fore-sights 
of  equal  length. 

When  stakes  are  set  at  regular  intervals,  there  is  no 
difficulty  in  determining  the  length  of  sights,  and  mak- 
ing them  equal ;  and  in  other  cases  the  distance  can  be 
determined  by  stepping.  When  extreme  accuracy  is 
desired,  the  rod-man  approximates  the  distance  by 
stepping,  and  the  instrument-man  measures  it  by  the 
principle  of  the  stadia  (Chap.  X).  All  levels  should  be 
provided  with  two  extra  horizontal  cross  hairs  for  this 
purpose. 

In  the  precise  leveling  on  the  U.  S.  Lake  Survey,  and 
in  surveys  made  under  the  direction  of  the  Mississippi 
River  Commission,  as  well  as  in  the  Swiss  levels  of  pre- 
cision, the  difference  between  corresponding  back-sight 
and  fore-sight  is  not  allowed  to  exceed  10  meters  (33 
feet). 

In  ascending  or  descending  a  hill,  it  is  nearly  impossi- 
ble, and  always  very  tedious,  to  make  the  back-sight 
and  fore-sight  equal.  As  the  rod  is  about  twice  as  high 
as  the  instrument,  the  down  hill  sight  will  be  about 
twice  the  length  of  the  up-hill  one.  When  the  ground 
renders  sights  of  unequal  length  unavoidable,  keep 
notes  of  the  distance;  and,  as  soon  as  possible,  take 
sights  with  corresponding  inequalities  in  the  contrary 
direction.  When  approaching  a  long  incline  make  part 
of  this  compensation  in  advance. 

327.  Reciprocal  Leveling.  In  crossing  a  river,  it  is 
absolutely  necessary  that  the  back-sight  and  fore-sight 
should  differ  considerably  ;  and  there  are  other  some- 
what similar  cases  in  which  they  can  not  be  made  even 
approximately  equal.  In  such  instances  the  principles 


ART.  7]  USING    THE    LEVEL.  289 

of  reciprocal  leveling  are  applicable.*  The  method  of 
procedure  is  as  follows  : 

Establish  a  bench  upon  both  sides  of  the  river  and 
determine  the  difference  of  level  ;  move  the  instrument 
to  the  other  side,  and  re-determine  the  difference  of 
level.  If  the  sights  are  taken  in  quick  succession,  the 
mean  of  the  two  results  is  the  true  difference  of  level. 
Simultaneous  observations  with  two  instruments  would 
be  still  better. 

The  instructions  on  the  U.  S.  Lake  Survey  were  to  (i) 
read  upon  a  bench  on  the  nearer  shore  with  the  tele- 
scope normal  and  also  inverted,  (2)  read  upon  the 
bench  on  the  farther  shore  five  times  with  the  telescope 
normal  and  five  times  with  it  inverted,  and  (3)  read 
upon  the  nearer  bench  as  before.  The  rod  on  the 
farther  shore  is  provided  with  a  target  from  6  inches  to 
i  foot  square,  according  to  the  distance.  The  line  of 
sight  should  pass  at  least  10  or  12  feet  above  the  water 
to  eliminate  abnormal  refraction  (§  318).  The  observa- 
tions must  be  corrected  for  curvature  and  refraction 
(§319).  With  a  single  Kern  level  (§  261),  this  process 
has  given  for  a  river  half  a  mile  wide  (the  Ohio,  at 
Cairo,  111.)  five  results  the  mean  of  which  had  a  proba- 
ble error  (App.  Ill,  §  2)  of  0.5  mm.  (0.002  ft.  nearly). f 
Of  course  this  must  be  regarded  as  an  exceptionally 
accurate  result,  but  it  gives  an  idea  of  the  practice  in 
such  matters.  In  another  case,  sixteen  simultaneous 
observations  on  each  side  gave  a  probable  error  of 
i  mm.  (0.003  ft-)  f°r  a  river  crossing  600  meters  (2,000 
ft.)  wide. 

*  The  surface  of  water  can  not  be  assumed  to  be  level  except  (i)  where 
there  is  no  current  or  wind,  or  (2)  where  the  thread  of  the  current  is  midway 
between  the  banks,  and  the  line  joining  the  two  points  of  observation  is  per- 
pendicular to  the  current,  and  there  is  no  wind. 

t  Report  of  the  Chief  of  Engineers,  U.  S.  A.,  for  1880,  Part  III,  p.  2432, 


2QO  SPIRIT    LEVELS.  [CHAP.   XI 

On  the  U.  S.  Coast  and  Geodetic  Survey,*  eleven 
sets  of  observations,  consisting  of  twelve  to  sixteen 
pointings  each,  made  at  four  different  hours  on  three 
different  days,  across  a  river  2,200  feet  wide,  gave  a 
probable  error  of  1.9  mm.  for  the  mean  of  each  set. 
In  the  above  observations  the  line  of  sight  was  about 
12  feet  above  the  water;  but  an  equal  number  of  ob- 
servations, made  at  the  same  time  as  the  above,  with 
the  line  of  sight  5  feet  above  the  surface,  gave  a  result 
for  the  difference  of  level  between  the  two  bench  marks 
on  opposite  sides  of  the  river  differing  from  the  mean 
of  the  above  observations  by  23  mm.  This  shows  the 
uncertainty  of  such  observations,  even  though  the  re- 
sults may  not  disagree  among  themselves. 

328.  If  the   line   to  be  leveled  passes  over  a  stream 
with  steep  high  banks,  or  over  a  narrow  deep  gorge  or 
valley,  establish  a  turning  point  on  the  farther  side  by 
reciprocal   leveling.     Then,   to   find    the    depth    of    the 
opening,  level  down  the  bank  without  much   regard  to 
equality  in  length  of  sights,  or  other  refinements.    This 
will  usually  be  all  that  is  necessary,  and  is  much  quicker 
done  than  leveling  down  one  bank  and  up  the  other. 

329.  Bench   Marks.      These    are    permanent    objects, 
natural  or  artificial,  whose  heights  are  determined  and 
recorded  for  future  reference.     Any  object   not   easily 
disturbed,  and  easily  described  and  found,  may  be  used 
as  a  bench  mark,  as,  for  example,  the  highest  point  of  a 
bowlder,  a  nail  in  the  root  of  a  prominent  tree,  a  stone 
door-sill,  etc.     A  stake   driven    to    the   surface    of    the 
ground,  with  another   projecting  above  the  ground  to 
mark    its  position,  is    frequently  used   in   railroad   and 
drainage  surveying  ;   but  it  is  not  very  reliable  at  best, 
and  is  entirely  unreliable  after  having  stood  over  winter, 
owing  to  the  "  heaving  of  the  frost."     The  precise  Joca- 

*  Annual  report  for  1879,  pp.  212-13. 


ART.  7]  USING    THE    LEVEL.  291 

tion  and  description  of  every  bench  should  be  given 
very  fully  and  definitely  in  the  "  remarks  "  column  of 
the  field  notes.  The  description  should  be  such  that  an 
entire  stranger  could  find  the  bench  by  the  aid  of  the 
notes  alone.  For  convenience,  the  benches  are  num- 
bered, the  number,  and  the  fact  that  it  is  a  bench,  being 
marked  on  or  near  it. 

Bench  marks  should  be  established  at  frequent  inter- 
vals along  the  line.  They  serve  as  points  of  beginning 
in  case  of  accident, — as,  for  example,  losing  a  turning 
point, — and  are  points  on  which  to  check  in  case  the 
line  is  re-run.  They  are  usually  put  in  at  each  mile 
and  half-mile  from  the  beginning  of  the  line  ;  and  they 
should  also  be  put  in  at  all  places  where  they  may  be 
necessary  or  convenient — as,  for  instance,  near  where 
the  line  crosses  a  prominent  road,  on  both  sides  of  a 
river  crossed,  near  the  top  or  bottom  of  a  high  hill 
crossed,  etc. 

330.  CARE.  For  remarks  on  the  care  of  instruments, 
which  are  applicable  to  the  level,  see  §  95  and  §  148. 


CHAPTER  XII. 
BAROMETERS. 

332.  THE  difference  of  level  of  two  places  may  be  de- 
termined by  ascertaining  the  difference  in  the  height  of 
the  atmosphere  above  the  places.  This  may  be  found 
in  any  of  three  ways;  viz.,  (i)  by  determining  how  high 
a  column  of  mercury,  or  other  liquid,  the  column  of  air 
above  it  will  balance,  (2)  by  finding  the  pressure  it 
will  exert  against  an  elastic  box  from  which  the  air  has 
been  exhausted,  or  (3)  by  observing  the  temperature  at 
which  a  liquid  boils,  i.e.,  by  observing  the  temperature 
at  which  the  pressure  of  the  atmosphere  just  balances 
the  tension  of  the  vapor.  There  are  then  three  slightly 
different  methods  of  barometric  leveling  according  as 
the  instrument  used  is  a  mercurial  barometer,  an  ane- 
roid barometer,  or  a  thermo-barometer  or  boiling-point 
apparatus.  As  the  thermo-barometer  has  been  super- 
seded by  the  aneroid,  only  the  methods  of  leveling  by 
the  mercurial  and  the  aneroid  barometers  will  be  con- 
sidered here. 

Barometric  leveling  is  specially  adapted  to  finding  the 
difference  of  level  between  points  at  considerable  hori- 
zontal or  vertical  distance  apart.  Under  these  condi- 
tions, it  is  the  most  expeditious,  but  the  least  accurate,  of 
any  of  the  methods  of  leveling.  It  is  very  valuable  in 
making  geographical  surveys  of  large  areas  for  deter- 
mining the  elevation  of  stations  to  be  occupied  by  the 
topographer.  It  is  also  well  suited  to  making  a  recon- 
noissance  for  a  railroad  or  for  a  scheme  of  triangulation. 

292 


ART.    l]  MERCURIAL    BAROMETER.  293 


ART.  1.     THE  MERCURIAL  BAROMETER. 

333.  CONSTRUCTION.  There  are  two  kinds  of  mercurial 
barometers,  the  cistern  and  the  siphon.  The  former  is 
the  better  and  more  reliable  for  hypsometrical  purposes. 
The  general  form  of  the  cistern  barometer  needs  no 
description.  Fig.  80,  page  294,  shows  some  of  the  details 
of  Green's  barometer,  which  is  generally  considered  one 
of  the  best.  The  right-hand  portion  of  Fig.  80  shows 
the  cistern  and  the  details  at  the. lower  end  of  the  in- 
strument, and  the  left-hand  portion  the  vernier  and 
scale  at  the  upper  end  of  the  mercury  column. 

The  cistern  consists  of  a  glass  cylinder  Ft  which  allows 
the  surface  of  the  mercury  to  be  seen,  and  a  top  plate 
G,  through  the  neck  of  which  the  barometer  tube  / 
passes,  and  to  which  it  is  fastened  by  a  piece  of  kid 
leather,  making  a  strong  but  flexible  joint.  To  this 
plate  is  attached  also  a  small  ivory  point,  h,  the  extrem- 
ity of  which  marks  the  commencement  or  zero  of  the 
scale.  The  lower  part,  containing  the  mercury  into 
which  the  end  of  the  tube,  t,  is  plunged,  is  formed 
of  two  parts,  i  and  /,  held  together  by  four  screws 
and  two  divided  rings,  /  and  m.  To  the  lower  piece,/, 
is  fastened  the  flexible  bag  N,  made  of  kid  leather,  fur- 
nished in  the  middle  with  a  socket,  /£,  which  rests  on  the 
end  of  the  adjusting  screw  O.  These  parts,  with  the 
glass  cylinder,  F,  are  clamped  to  the  flange,  B,  by  means 
of  four  large  screws,  P,  and  the  ring,  J?.  On  the  ring,  R 
screws  the  cap,  S,  which  covers  the  lower  parts  of  the 
cistern  and  supports  at  the  end  the  adjusting  screw  O. 
G,  i,j,  and  k,  are  of  boxwood  ;  the  other  parts  of  brass  or 
German  silver.  The  screw  O  serves  to  adjust  the  mer- 
cury to  the  ivory  point,  and  also,  by  raising  the  bag  so 
as  to  completely  fill  the  cistern  and  tube  with  mercury, 
to  put  the  instrument  in  condition  for  transportation. 


204 


BAROMETERS. 


[CHAP.  XL 


The  milled-head  screw  D  moves,  by  means  of  a  rack 
and  pinion,  the  vernier  C,  so  as  to  bring  the  horizontal 


FIG.  80. — MERCURIAL  BAROMETER. 

line  just  below  C  level  with  the   top   of   the   mercury 
column. 

334.  CLEANING  THE  BAROMETER.  It  frequently  hap- 
pens that  the  mercury  in  the  cistern  becomes  so  dirty 
that  the  ivory  point,  or  its  reflection  in  the  mercury,  can 
no  longer  be  seen.  This  often  occurs  even  though  the 
barometer  is  in  good  condition  in  every  other  respect. 


ART.   l]  MERCURIAL    BAROMETER.  $9$ 

"The  instrument  can  be  taken  apart  and  cleaned  with 
safety  and  without  changing  in  the  slightest  degree  the 
zero  of  the  instrument.  Everything  used  in  the  opera- 
tion must  be  clean  and  dry.  Avoid  blowing  upon  any 
of  the  parts,  as  the  moisture  from  the  breath  is  in- 
jurious. 

"  Turn  up  the  adjusting  screw  at  the  bottom  until 
the  mercury  entirely  fills  the  tube,  carefully  invert, 
place  the  instrument  firmly  in  an  upright  position,  un- 
screw and  take  off  the  brass  casing  which  encloses  the 
wooden  and  leather  parts  of  the  cistern.  Remove  the 
screws,  and  lift  off  the  upper  wooden  piece  to  which  the 
bag  is  attached;  the  mercury  will  then  be  exposed.  By 
inclining  the  ins-trument  a  little,  a  portion  of  the  mer- 
cury in  the  cistern  may  be  poured  out  into  a  clean  ves- 
sel at  hand  to  receive  it,  when  the  end  of  the  tube  will 
be  exposed.  This  is  to  be  closed  by  the  gloved  hand, 
when  the  instrument  can  be  inverted,  the  cistern 
emptied,  and  the  tube  brought  again  to  the  upright 
position.  Great  care  must  be  taken  not  to  permit  any 
mercury  to  pass  out  of  the  tube.  The  long  screws 
which  fasten  the  glass  portion  of  the  cistern  to  the  other 
parts  can  then  be  taken  off,  the  various  parts  wiped  with 
a  clean  cloth  or  handkerchief,  and  restored  to  their 
former  position. 

*'  If  the  old  mercury  is  merely  dusty,  or  dimmed  by 
the  oxide,  the  cleaning  may  be  effected  by  straining  it 
through  chamois  leather,  or  through  a  funnel  with  a 
capillary  hole  at  the  end  of  a  size  to  admit  of  the  pas- 
sage of  but  a  small  thread  of  the  metal.  Such  a  funnel 
is  conveniently  made  of  letter  paper.  The  dust  will 
adhere  to  the  skin  or  paper,  and  the  filtered  mercury 
will  present  a  clean  and  bright  appearance.  If  chemi- 
cally impure,  it  should  be  rejected,  and  fresh,  clean 
mercury  used.  With  such  clean  mercury  the  cistern 
should  l>e  filled  as  nearly  full  as  possible,  the  wooden 


296  BAROMETERS.  fcHAP.  XII 

portions  put  together  and  securely  fastened  by  the 
screws  and  clamps,  the  brass  casing  screwed  on,  and 
the  screw  at  its  end  screwed  up.  The  instrument  can. 
then  be  inverted,  hung  up,  and  re-adjusted.  The  tube 
and  its  contents  having  been  undisturbed,  the  instrument 
should  read  the  same  as  before."  * 

With  the  instrument  before  the  operator,  these  in- 
structions are  easily  understood.  In  this  case,  as  in 
using  and  caring  for  any  instrument,  a  little  care  and  a 
thoughtful  inspection  of  the  method  of  construction  is 
worth  more  than  any  written  description.  If  a  little 
mercury  has  been  lost  during  the  operation,  and  there 
is  none  at  hand  to  replace  it,  no  serious  harm  has  been 
done;  but  if  much  is  lost,  the  open  end  of  the  tube  may 
become  exposed  in  inverting  the  instrument,  in  which 
case  air  may  enter. 

335.  FILLING  THE  BAROMETER.  It  is  no  slight  matter 
to  properly  fill  a  barometer.  It  can  best  be  done  by 
the  manufacturer,  who  has  all  the  facilities;  but  as  it 
is  sometimes  necessary  for  the  observer  to  re-fill  his 
barometer,  the  following  hints  are  given.  Tubes  require 
re-filling  owing  to  the  breakage  of  the  glass  or  to  the 
entrance  of  a  bubble  of  air. 

The  mercury  should  be  chemically  pure  and  free  from 
oxide,  otherwise  it  will  adhere  to  the  glass  and  tarnish 
it.  Moreover,  if  it  is  not  pure,  the  height  of  the  baro- 
metric column  will  not  be  correct.  No  mercury  should 
be  used  except  that  which  has  been  purified  by  distilla- 
tion. For  the  best  results,  the  mercury  should  be  boiled 
in  the  tube  to  expel  moisture  and  air;  but  this  can  not 
always  be  done,  and  fair  results  may  be  obtained  with- 
out boiling. 

In  extended  barometric  operations  in  the  field,  a  sup- 
ply of  extra  tubes  is  carried,  to  be  used  in  case  a  tube  is 

*  On  the  Use  of  the  Barometer  on  Surveys  and  Reconnoissances,  Maj.  R. 
S.  Williamson,  U.  S.  A.,  New  York  City,  1868,  pp.  136  and  137. 


ART.   l]  MERCURIAL    BAROMETER.  297 

broken.  These  tubes  should  be  drawn  out  so  as  to  be 
a  little  longer  than  the)7  are  required  to  be  when  fitted 
into  the  barometer.  The  open  end  should  be  cut  off  to 
such  a  length  that  it  shall  always  be  immersed  and  yet 
not  interfere  with  the  rise  of  the  lower  part  of  the  cis- 
tern. When  the  instrument  is  finally  put  together,  the 
cork  in  the  upper  end  of  the  brass  case  should  be  ad- 
justed so  as  to  hold  the  closed  end  of  the  tube  firmly. 

336.  By  Boiling.     "  To  fill  a  tube  by  boiling,  an  alco- 
hol lamp  is  needed,  although  it  can  be  done  over  a  char- 
coal fire.    The  lamp  being  filled  and  put  in  order,  begin 
to  fill   the    tube  by   pouring  in   through   the  funnel  as 
much  warm  mercury  as  will  occupy  about  5  inches;  then, 
holding  the  tube  with  both  hands  above  the  mercury, 
heat  it  gently,  and   let   the  mercury  boil  from    the  sur- 
face downward  to  the  end  of  the  tube,  and  then  back 
again,  chasing  all  of   the    bubbles  of   air   upward.      A 
little  practice  will  make  this  easy,  the  tube  being  held 
a  little  inclined  from  the  horizontal,  and  constantly  and 
rapidly  revolved,  always  in   the  same  direction,  so  that 
every  portion  of  the  metal  maybe  heated  gradually  and 
uniformly.     After  this  has  been   done,  let  the  tube  cool 
sufficiently  to  admit  of  its  being  held  by  the  gloved  hand, 
and  then  pour  in  enough  warm  mercury  to  occupy  sev- 
eral inches  more  of  the  tube,  which    may  now  be  held 
with  both   hands,  one  above   and   the   other  below  the 
heated  portion.    After  boiling  this  thoroughly  free  from 
air,  repeat  the  same  operation  with  more  mercury  added, 
until  the  tube  is  filled  to  the  end.     With  care  and  prac- 
tice the  mercury  may  be  boiled  entirely  free  from  air 
up  to  within  an  inch  or  less  of  the  end  of  the  tube.     A 
tube  filled  in   this  way  may  have,  in  every  respect,  as 
perfect   a  vacuum  as    one    prepared  by  a  professional 
instrument  maker."  * 

337.  Without  Boiling.     The  glass  tube,  which  should 

*  Williamson's  On  the  Barometer,  p.  138. 


298  BAROMETERS.  [CHAP.  XII 

be  clean  and  dry,  must  have  its  open  end  ground  straight 
and  smooth,  so  that  it  can  be  closed  air-tight  with  the 
finger,  which  should  be  covered  with  a  piece  of  chamois 
or  kid  skin.  Warm  well  both  mercury  and  glass  tube, 
and  through  a  clean  paper  funnel  with  a  very  small  hole 
(about  ^Q-  of  an  inch)  below,  filter  in  the  mercury  to 
within  one  fourth  of  an  inch  of  the  top.  Shut  up  the 
end  and  turn  the  tube  horizontal,  when  the  mercury  will 
form  a  bubble  which  can  be  made  to  run  from  end  to 
end  by  a  change  of  inclination,  and  which  will  gather 
all  the  small  air  bubbles  that  adhered  to  the  inside  of 
the  glass  tube  during  filling.  Let  this  bubble,  which 
has  grown  somewhat  larger,  pass  to  the  open  end;  then 
fill  the  tube  completely  with  mercury,  and  shut  it  tightly. 
Next  reverse  the  tube  over  a  basin,  when,  by  slightly 
relieving  the  pressure  against  the  end,  the  weight  of 
the  column  of  mercury  will  force  some  out,  forming  a 
vacuum  above,  which  ought  not  to  exceed  one  half  an 
inch.  Closing  up  again  tightly,  let  this  vacuum  bubble 
traverse  the  length  of  the  tube  on  the  several  sides,  that 
it  may  absorb  those  minute  portions  of  air  that  were 
not  drawn  out  at  the  first  gathering,  and  which  are  now 
greatly  expanded  from  removed  atmospheric  pressure. 
The  complete  absence  of  air  is  easily  recognized  by  the 
sharp  concussions  with  which  the  column  beats  against 
the  sealed  end,  when  the  tube  is  held  horizontally  and 
slightly  moved. 

"  A  barometer  filled  without  boiling  will  probably 
read  lower  by  a  few  thousandths  than  if  the  tube  had 
been  boiled;  but  in  a  stationary  barometer  its  error  will 
probably  not  soon  change,  and  carrying  on  horseback 
will  be  apt  to  improve  it  rather  than  otherwise,  as  it  is 
then  carried  with  the  cistern  uppermost  and  the  bubbles 
will  be  jolted  toward  the  open  end.  If  possible  it  should 
be  compared  with  a  standard  barometer."* 

*  Williamson's  On  the  Barometer,  p.  140. 


ART.    l]  MERCURIAL    BAROMETER.  2!)9 

338.  READING  THE  BAKOMETEK.  Read  the  attached 
thermometer  first.  It  is  more  sensitive  than  the  barom- 
eter, and  is  affected  by  the  heat  of  the  body,  while  the 
barometer  is  not  so  affected.  The  thermometer  should 
be  read  as  closely  as  possible,  for  a  difference  of  i°  F.  is 
equivalent  to  about  3  feet  in  altitude.  Parallax  should 
be  carefully  avoided  in  making  this  reading. 

Then,  by  means  of  the  adjusting  screw  at  the  lower 
end  of  the  instrument  (<?,  Fig.  80,  page  294),  bring  the 
ivory  point  just  to  the  mercury  in  the  cistern.  If  there 
is  a  line  of  light  visible  between  the  point  and  mercury, 
the  mercury  in  the  cistern  is  too  low;  and  if  the  point 
makes  a  depression,  the  mercury  in  the  cistern  is  too 
high.  If  neither  a  line  of  light  nor  a  depression  can  be 
seen,  the  adjustment  has  been  correctly  made.  It  is 
usually  best  to  lower  the  screw  till  a  distinct  line  of 
light  can  be  seen,  and  then  gradually  raise  it  until  the 
light  disappears.  When  the  mercury  is  bright,  a  shadow 
of  the  point  can  be  seen,  and  if  the  shadow  and  'the 
point  itself  form  a  continuous  unbroken  line,  the  screw 
at  the  bottom  of  the  cistern  has  been  properly  ad- 
justed. Before  making  the  final  adjustment,  tap  the 
barometer  a  little  just  above  the  cistern,  to  destroy  the 
adhesion  of  the  metal  to  the  glass.  Complete  the  con- 
tact of  the  mercury  and  the  ivory  point,  at  the  same  time 
being  certain  that  the  barometer  hangs  freely,  /'.*.,  ver- 
tically. 

Next,  tap  the  barometer  gently  in  the  neighborhood 
of  the  top  of  the  mercury  column,  to  destroy  the  adhe- 
sion of  the  mercury.  This  is  very  important,  since 
raising  or  lowering  the  mercury  in  the  previous  opera- 
tion materially  affects  the  form  of  the  upper  surface. 
Then  take  hold  lightly  of  the  brass  casing  of  the  barom- 
eter at  a  distance  from  the  attached  thermometer,  that 
neither  the  case  nor  the  thermometer  may  be  unneces- 
sarily heated,  and  by  means  of  the  milled-head  screw 


300  BAROMETERS.  [CHAP.  XII 

near  the  middle  of  the  tube  (Z>,  Fig.  80),  bring  the  front 
and  back  edge  of  the  vernier  into  the  same  horizontal 
plane  with  the  top  of  the  mercury  in  the  tube,  and 
remove  the  hand  to  allow  the  instrument  to  hang 
vertically.  Move  the  eye  about,  and  if,  in  any  position, 
a  line  of  light  can  be  seen  between  the  mercury  and  the 
vernier,  the  latter  must  be  moved  down;  if  there  is  no 
line  of  light  and  a  portion  of  the  meniscusjs  obscured,  the 
vernier  must  be  moved  up.  As  the  top  of  the  column 
is  more  or  less  convex,  when  the  adjustment  is  correctly 
made  a  small  place  is  obscured  in  the  center  while  the 
light  is  seen  on  either  side. 

Finally,  having  adjusted  the  instrument  as  above,  it 
may  be  read  at  leisure.  On  the  best  barometers  the 
scale  is  divided  to  inches,  tenths,  and  half-tenths,  and 
the  vernier  reads  to  one  twenty-fifth  of  a  half-tenth 
(sV  X  0.05),  or  two  thousandths  (0.002)  of  an  inch.  (See 
Fig.  15,  page  69.) 

339.  TRANSPORTING  THE  BAROMETER.  "  In  transporting 

a  barometer,  even  across  a  room,  it  should  be  screwed 
up,  and  carried  with  its  cistern  uppermost.  For  travel- 
ing it  should  be  provided  with  a  wooden  and  leather 
case.  On  steam  boats  or  steam  cars  it  should  be  hung 
up  by  a  hook.  In  wheeled  vehicles  it  should  be  carried 
by  hand,  supported  by  a  strap  over  the  shoulder  or  held 
upright  between  the  legs;  but  it  should  not  be  allowed 
to  rest  on  the  floor  of  the  carriage,  for  a  sudden  jolt 
might  break  the  tube.  If  carried  on  horseback  it 
should  be  strapped  over  the  shoulder  of  the  rider,  where 
it  is  not  likely  to  be  injured  unless  the  animal  is  sub- 
ject to  a  sudden  change  of  gait.  When  about  to  be 
used,  it  should  be  taken  from  its  case  while  screwed 
up,  gently  inverted  and  hung  up,  when  it  can  be  un- 
screwed. While  it  has  its  cistern  uppermost  the  tube  is 
full,  is  one  solid  mass  of  metal  and  glass,  and  not  easily 
injured  ;  but  when  hung  up,  a  sudden  jolt  might  send 


ART.  2]  ANEROID   BAROMETERS.  3<M 

a  bubble  of  air  into  the  vacuum  at  the  upper  end  of 
the  tube,  and  the  instrument  would  be  useless  until  re- 
filled." * 


ART.  2.     ANEROID  'BAROMETERS. 

340.  The  aneroid  barometer  consists  of  a  cylindrical 
metallic  box,  from  which  the  air  is  exhausted,  having  a 
thin  corrugated  metal  top  which  readily  yields  to  alter- 
ations in  the   pressure   of   the   atmosphere.     When  the 
pressure  increases,  the  top  is   pressed  inwards;  when  it 
decreases,  the  elasticity  of  the  lid  tends  to  move  it  in 
the  opposite  direction.     There  are  two  general  forms  of 
aneroids,  according  to  the  method  employed  for  read- 
ing the  movements  of  the  top  of^  the  vacuum  chamber. 
In   the  common  form  these   motions  are  transmitted  by 
delicate  multiplying   levers   to  an    index  which    moves 
over    a    scale.     In  the    Goldschmid   aneroid  the  move- 
ments of  the  top  of  the  vacuum  box  cause  an  index  to 
move  through  the  field  of  a  micrometer  microscope  by 
which  the  movement  is  measured. 

341.  COMMON  OK  VIDI  ANEROID.    Fig.   81,  page  302, 

shows  the  mechanism  of  the  ordinary  form  of  aneroid 
barometer.  The  outside  case  and  the  graduated  dial 
are  not  shown.  The  movement  of  the  top  of  the 
vacuum  chamber,  M,  is  communicated  to  the  index 
through  the  levers  /,  ;//,  and  r,  and  the  watch  chain  S 
which  winds  around  the  axis  carrying  the  index.  The 
broad  curved  spring  R  keeps  the  lever/  in  contact  with 
the  post  on  the  top  of  the  vacuum  chamber. 

There  are  several  modifications  of  this  form  which 
differ  in  the  mechanism  employed  to  multiply  the  linear 
motion  of  the  end  of  the  vacuous  box,  and  to  convert  it 
into  angular  motion.  A  spring  is  sometimes  inserted 

*  Williamson's  On  the  Barometer,  p.  134. 


302  BAROMETERS.  [CHAP.  XII 

between  the  two  ends  of  the  vacuum  chamber  to  rein- 
force the  elasticity  of  the  corrugated  ends. 

Sometimes  the  vacuous  box  is  not  entirely  exhausted, 
the  claim  being  that  the  enclosed  air  renders  the  indi- 


FIG.  81. — COMMON  ANEROID  BAROMETER. 

cations  of  the  instrument  independent  of  changes  of 
temperature.  Such  aneroids  are  said  to  be  compen- 
sated; but  the  theory  is  incorrect,  and  some  "compen- 
sated "  aneroids  are  more  affected  by  changes  of  tem- 
perature than  uncompensated  ones  (see  §  343,  par.  2). 
An  aneroid  should  have  a  thermometer  attached  to  it. 

The  mechanism  shown  in  Fig.  81  is  enclosed  in  a 
brass  or  silver  case,  which  varies  from  2  to  6  inches  in 
diameter,  and  from  0.5  to  3  inches  in  thickness.  The 
smaller  sizes  are  usually  made  of  silver,  in  the  form  of 
a  watch,  and  are  known  as  pocket  aneroids.  The 
larger  ones  look  like  a  short  brass  cylinder,  and  are 


ART.   2]  ANEROID    BAROMETERS.  303 

carried  in  a  leather  case  supported  by  a  strap  over  the 
shoulder. 

342.  The  face  of  the  instrument  has  a  scale  of  inches, 
and  sometimes  also  a  scale  of  elevations.     The  former 
is  graduated  empirically  by  comparing  its  indications 
under    different    pressures    with    those   of  a   mercurial 
barometer.       The    scale    is    marked    to    correspond    to 
inches  of    the  ordinary   barometer  column,   the   inches 
being  divided  into  tenths,  and  the  tenths  usually  into 
four  parts.     At  the  back  of   the   instrument  is  a  little 
screw  which  presses  against  one   end  of  the  exhausted 
box;  and  by  turning  this  screw  the  index  can  be  moved 
over  the  scale,  and  the   instrument   may  thus  be  made 
to  agree  at  any  time  with  a  standard  mercurial  barom- 
eter. 

The  altitude  scale  is  obtained  by  converting  the  scale 
of  inches  into  elevations  by  the  use  of  some  barometric 
formula  (see  Art.  4),  and  engraving  the  results  upon 
the  face,  adjacent  to  the  scale  of  inches.  The  altitude 
scale  is  at  best  of  doubtful  utility  (see  §  403). 

343.  Defects.     The  aneroid   is  a  very  convenient  in- 
strument, and    where    nice   readings  are    not    required 
it  does  very  well;    but  for  accurate  hypsometrical  re- 
sults it  is  an  inferior  instrument.     It  has  the  following 
defects: 

1.  The  elasticity  of  the  corrugated  top  of  the  vacuum 
chamber  is   affected   by  repeated  changes  in   pressure. 
This  will  produce  error  in  the  scale  readings. 

2.  It  is   usually  claimed   that,  in   consequence   of  not 
completely  exhausting  the  vacuum  box,  the  indications 
of    the    aneroid   become   independent  of    the   effect   of 
changes  of  temperature  of    the  instrument.     The  best 
that  can   be  hoped  is  that  for  small  changes  the   tem- 
perature correction  is   less   than  the  error  of   observa- 
tion.    In  instruments  compensated  for  temperature,  the 
effect  of  a  change  is  sometimes  the  same  as   that  in  the 


304  BAROMETERS.  [CHAP.  XII 

mercurial  barometer,  and  sometimes  the  reverse.  The 
effect  of  the  temperature  on  any  particular  instrument 
can  be  determined  only  by  trial. 

3.  The  different  spaces  on  the  scale  of  inches  are  sel- 
dom correct  relative   to  each  other,  owing  probably  to 
errors  of  observation   and  graduation,  and   possibly  to 
differences  of  temperature   and    changes   in    elasticity. 
As  a  matter  of  fact,  the  scale  is  often  only  a  scale  of 
equal    parts.     The    barometer    scale    is   more  accurate 
than  the  elevation    scale,  since  the  latter  has  all  the  in- 
accuracies due  to  the  formula  by  which  it  is  graduated 
(§   403),  in   addition    to   those   of  the  instrument  itself. 
Before  using  the  aneroid,  it  should  be  compared  with  a 
mercurial  column  under  an  air-pump,  to  determine  the 
errors  of  its  scale  for  different  temperatures  and  press- 
ures. 

4.  The  weight  of  the  machine  affects  its  indication, 
i.e.,  the  reading  of  the  aneroid  will  differ  when  held  in 
different  positions.     In  the  best  instruments  this  differ- 
ence is  sometimes  as  much  as  0.008  of  an   inch,  corre- 
sponding to   a  difference  of  elevation  of  about  8  feet. 
Usually  the  reading  is  higher  with  the  dial  is  horizontal 
(face  uppermost),  than  when  it  is  vertical. 

5.  Like  all  combinations  of  levers,  screws,  and  springs, 
the    aneroid    is    liable    to    continual   shifting    of    parts, 
when    subject    to    the    jars    and    jolts    encountered  in 
transportation   and  in    use.     The    only   remedy   is  fre- 
quent comparisons  with  a  mercurial  barometer. 

6.  The  aneroid  is    deficient  in    precision,   since    the 
least  reading    is  0.025   °f    an   inch,   which  corresponds 
nearly  to  25  feet  of  elevation. 

7.  With  .most  aneroids  the  spring  ceases  to  act  after 
the    pressure  has  been  lowered  somewhat,  i.e.,  the  in- 
strument runs  down.       Before    using    it,    experiments 
should  be  made  to  determine  the  range  of  pressure  to 
which  it  may  be   exposed   before    the  spring  ceases   to 


ART.   2]  ANEROID    BAROMETERS.  305 

act.  In  case  an  aneroid  is  to  be  used  in  an  elevated 
region,  if  there  is  a  mercurial  barometer  with  the  party, 
screw  up  the  anerord  until  the  spring  acts  well,  and  set 
the  instrument  by  the  mercurial  barometer  so  that  there 
shall  be  a  considerable  difference,  say  2  or  3  inches,  be- 
tween them.  Of  course  this  difference  must  be  added 
to  each  reading  of  the  aneroid. 

344.  "  With    all  these    defects  a  good  aneroid  is  of 
much    assistance    on    a    survey    or    reconnoissance    in 
mountainous    districts,   on    side    trips  of  one,  or  even 
several,  day's  duration,  when  the  instrument  has  been 
previously  compared  with  a  standard  mercurial  barom- 
eter at  various  temperatures  and  in  different  elevations, 
and  proper  tables  of  corrections  made.      It   should    be 
compared  before  and  after  it  is  used  in  that  way,  to  see 
if  the  zero  has  not  changed  in  the  meantime,  and  if  the 
agreements  are   satisfactory   the  results  can   be   relied 
upon.     It  is  evidently  important  that  there  should  be 
a  good  attached  thermometer."  * 

345.  Reading  the   Aneroid.      In    measuring    heights 
with  the  aneroid  barometer,  it  should  be  shielded  from 
the  direct   rays  of  the   sun,  and  care  should  be  taken 
that  it  is   not  unnecessarily   influenced  by  the   heat  of 
the   body   of    the   observer.      If    the   instrument   has   a 
thermometer  attached,  that  should  be  read  first,  as  it  is 
liable  to  be  affected  by  the  heat  of  the  hand  of  the  ob- 
server. 

The  barometer  should  be  held  in  the  same  position 
for  both  observations — preferably  with  the  face  hori- 
zontal; and  should  be  tapped  gently  with  the  finger 
just  before  taking  the  reading.  Considerable  care  i? 
required  to  determine  exactly  where  the  index  pointy 
Some  instruments  are  provided  with  a  small  lens  for 
this  purpose  (see  last  paragraph  on  page  94).  Without 

*  Williamson's  On  the  Barometer,  p.  133. 


BAROMETERS. 


[CHAP,  xii 


the  lens,  the  chief  error  is  due  to  parallax.  Of  course, 
the  index  should  move  very  close  to  the  graduation 
and  yet  not  touch  the  face  of  the  instrument. 

346.  GOLDSCHMID  ANEROID.  The  common  aneroid  was 
invented  about  the  beginning  of  this  century,  but  was 
first  made  of  a  serviceable  form  by  Vidi,  of  Paris,  in 


FIG.  82.— GOLDSCHMID  ANEROID. 

1847.  The  defects  of  its  complicated  system  of  mul- 
tiplying levers  have  long  been  recognized;  and  as  early 
as  1857,  Goldschmid  designed  a  form  of  aneroid  in 
which  he  dispensed  with  the  transmitting  and  multiply- 
ing mechanism  of  the  Vidi  form. 

Figs.  82  and  83,  pages  306  and  307,  are  two  views  of  one 
of  the  latest  forms  of  the  Goldschmid  aneroid.     Fig.  82 


ART.  2] 


ANEROID    BAROMETERS. 


307 


is  an  external  view,  showing  the  microscope  Z,  and  the 
micrometer  M ';  and  Fig.  83  is  a  section  showing  the 
compound  vacuum  chamber.  Obviously,  the  greater 
the  number  of  boxes,  the  larger  the  motion  of  the  index 
a.  The  relative  position  of  the  movable  index  a  and 
the  fixed  point  of  reference  £,  is  observed  by  the  tele- 
scope Z,  and  the  distance  is  measured  by  the  micrometer 
M.  The  instrument  is  very  delicate  in  its  indications, 


FIG.  83. — SECTION  OF  GOLDSCHMID  ANEROID. 

but  is  liable  to  serious  disarrangement  by  ordinary 
handling.  Different  manufacturers  make  slightly  dif- 
ferent forms  of  the  Goldschmid  type,  but  all  have  essen- 
tially the  same  defect,  i.e.,  are  not  able  to  stand  ordinary 
use. 

It  is  doubtful  if  there  is  any  advantage  in  an  aneroid 
as  complicated  as  that  shown  in  Figs.  82  and  83.  It 
seems  probable  that  no  form  can  be  devised  which 
shall  be  both  delicate  in  its  indications  and  able  to 
stand  rough  handling.  The  chief  advantage  of  the 


308  BAROMETERS.  [CHAP.  XII 

common  aneroid  is  its  portability,  combined  with  mod- 
erate accuracy.  The  mercurial  and  the  aneroid  barom- 
eters supplement  each  other  ;  the  first  is  delicate  and 
the  second  is  portable.  These  qualities  can  not  be  com- 
bined in  a  single  instrument,  nor  can  one  be  obtained 
more  delicate  or  more  reliable  than  the  mercurial 
barometer. 


ART.  3.     THE  OBSERVATIONS. 

347.  OUTLINE  OF  METHOD.     To  determine  a  difference 
of  elevation  with  the  barometer  it  is  necessary  to  find  at 
each  of  the  two  stations  (i)  the  reading  of  the  barometer, 
(2)  the  temperature  of  the  barometer,  and  (3)  the  tem- 
perature of  the  air.     Sometimes  observations  are  made 
to  determine   (4)   the  amount  of  watery  vapor  in   the 
atmosphere.       Inserting    these    data    in    a    barometric 
leveling  formula  (Art.  4)  and  reducing  gives  the  differ- 
ence of  elevation. 

Before  discussing  the  method  further,  it  is  necessary 
to  consider  the  errors  to  which  barometric  leveling  is 
liable. 

348.  SOURCES  OF  ERROR.     For  convenience  of  discus- 
sion, the  sources  of  error  will  be  considered  under  five 
heads ;    viz.,   (i)   gradient,   (2)    temperature   of  air,   (3) 
humidity,  (4)  instrumental  errors,  (5)  errors  of  observa- 
tion, and  (6)  effect  of  the  wind. 

349.  Gradient  Errors.     Let  A,  B,  and  C designate  three 
points  at  which  the  pressure  is  the  same.     The  plane 
passing  through  A,  B,  and  C  is  then  a  surface  of  equal 
pressure.     If  the  air  were  in  a  state  of  equilibrium,  this 
plane  would  be  level  ;  but  under  ordinary  conditions  it 
will  be  inclined  in  some  direction.     The  inclination   of 
this  surface  is  called  the  barometric  gradient. 

Instead  of  considering  only  three  points,  we  can  in 
imagination  project  through  the  air  a  surface  contain- 


ART.  3]  THE    OBSERVATIONS.  309 

ing  all  points  which  have  the  same  pressure.  If  the 
atmosphere  were  at  rest,  this  surface  would  be  a  hori- 
zontal plane  ;  but  under  the  actual  conditions,  it  is 
never  a  plane  and  is  ever  undulating.  For  small  areas 
under  ordinary  conditions,  this  surface  would  probably 
not  differ  much  from  a  plane. 

Conceive  another  surface  passed  through  all  points 
at  which  the  pressure  differs  from  the  preceding  one  by 
any  constant  quantity.  With  atmospheric  equilibrium 
two  such  surfaces  would  be  both  level  and  parallel,  but 
in  the  actual  case  they  are  neither  level  nor  parallel. 
When  widely  separated  surfaces  are  compared,  the 
variations  from  parallelism  are  often  so  great  that  their 
inclinations  above  the  same  locality  have  opposite  direc- 
tions. The  atmospheric  gradient  at  the  surface  of  the 
ground  may  therefore  differ  greatly  in  amount  and 
direction  from  the  simultaneous  gradient  at  a  consid- 
erable altitude  above  that  point. 

350.  That  the  atmosphere  is  not  in  static  equilibrium 
is  shown  by  its  being  continually  in  motion  and  also  by 
the  variation  in  the  height  of  the  barometer.  A  varia- 
tion of  the  barometric  pressure  indicates  a  change  in 
the  barometric  gradient,  and  a  gradient  or  a  change  in 
the  gradient  may  produce  an  error  in  the  result  ob- 
tained by  barometric  leveling.  For  example,  if  A  and 

B,  Fig.  84,  page  310,  are  two  stations  and   the   atmos- 
phere is  at  rest,  the  surface  of  equal  pressure,  BC,  is  a 
horizontal  plane,  and  AC  is  the  difference  of   elevation 
which  would  be  obtained   by  applying  any  one  of  the 
common  barometric  formulas.     If  the  air  is  not  in  static 
equilibrium,  the  pressure  at  A  will  be  greater  or  less 
than  before,  and  the  surface  of  equal  pressure  will  lie 
above  or  below  BC.     If  the   pressure  at  A  is  greater 
than  the  average,  the  surface  of  equal  pressure  is  above 

C,  say  at  E,  and  AE  is  the  corresponding  difference  of 
elevation  ;  and  similarly,  if  BD  is  the  surface  of  equal 


BAROMETERS. 


[CHAP,  xii 


pressure,  AD  is  the  corresponding  difference  of  eleva- 
vation. 

The  problem  is  further  complicated  by  the  fact  that 
the  air  above  B  also  is  in  a  state  of  oscillation.     If  the 


FIG.  84. 


variations  in  pressure  at  the  two  stations  were  simulta- 
neous and  alike  in  amount,  no  error  would  be  pro- 
duced by  the  barometric  gradient  ;  but  these  condi- 
tions are  seldom  or  never  realized,  and  therefore  there 
is  always  a  possibility  of  error  in  barometric  leveling, 
owing  to  the  barometric  gradient  and  also  to  a  change 
in  the  gradient  between  the  observations  at  the  two 
stations. 

351.  There  are  four  classes  of  barometric  gradient  ; 
viz.,    (i)  diurnal,  (2)    annual,  (3)  non-periodic,  and   (4) 
permanent. 

352.  Diurnal  Gradient.    It  is  a  fact  familiar  to  meteor- 
ologists that  the  pressure  of  the  air  everywhere  under- 
goes  a  daily  oscillation.     The  gradient  introduced  by 
this  daily  change  is  called  the  diurnal  gradient.     The 
pressure  has   two  maxima  and  two  minima  which  are 
easily  distinguishable.     Near  the  sea-level  the  barometer 
attains  its  first  maximum   about  9  or  10   A.M.     In   the 
afternoon  there  is  a  minimum  about  3  to  5  P.M.     It  then 
rises  until  10  or  n  P.M.,  when  it  falls  again  until  about 


ART.  3]  THE    OBSERVATIONS.  3ft 

4  A.M.,  and  again  rises  to  attain  its  forenoon  maximum. 
The  maxima  occur  when  the  temperature  is  about  the 
mean  of  the  day,  and  the  minima  when  the  temperature 
is  at  the  highest  and  lowest  respectively.  The  day 
fluctuations  are  the  larger. 

The  daily  oscillation  is  subject  to  variations  in  char- 
acter and  magnitude.  It  is  greater  in  summer  than  in 
winter  ;  and  is  greatest  at  the  equator  and  diminishes 
toward  the  poles,  but  is  not  the  same  for  all  places  of  the 
same  latitude.  Within  the  United  States  it  varies  be- 
tween 40  and  120  thousandths  of  an  inch.  Changes  of 
altitude  often  cause  a  marked  increase  in  the  amount 
of  the  diurnal  oscillation.  The  difference  which  per- 
tains to  latitude  does  not  materially  affect  the  ordinary 
hypsometric  problem,  but  the  difference  depending  on 
the  altitude  has  a  very  important  effect. 

A  change  of  i  thousandth  of  an  inch  in  the  height 
of  the  barometer  corresponds  to  a  difference  of  eleva- 
tion of  from  0.8  to  i.o  foot.  Therefore,  if  an  observa- 
tion were  made  at  one  station  at  about  10  A.M.,  and  at  a 
second  station  at  about  3  P.M.,  the  difference  of  elevation 
would  probably  be  from  40  to  120  feet  in  error. 

353.  Annual  Gradient.  The  annual  progress  of  the 
sun  from  tropic  to  tropic  throws  a  preponderance  of 
heat  first  on  one  side  of  the  equator  and  then  on  the 
other,  which  produces  an  annual  cycle  of  changes  in 
the  pressure  and  gives  rise  to  what  has  been  called  the 
annual  gradient.  The  amount  of  variation  in  the  baro- 
metric pressure  is  quite  small  near  the  equator,  but 
increases  rapidly  toward  the  poles.  The  mean  annual 
variation  in  the  United  States  ranges  from  120  to  200 
thousandths  of  an  inch,  although  the  variation  at  any 
particular  station,  or  for  any  one  year,  may  be  very 
much  greater.  For  interesting  diagrams  showing  the 
annual  variation  for  a  great  number  of  stations,  see 
Williamson's  On  the  Barometer,  pages  68-81. 


312  BAROMETERS.  [cHAP.  XIl 

354.  Non-periodic  Gradient.     In  addition  to  the  diurnal 
and  annual  variations  in  the  pressure,  there  are  others 
due  to  the  same  general  cause — the  heat  of  the   sun, — 
but   so  modified  by  the  local  conditions — topography, 
humidity,  winds,  storms,  etc., — as  to    make    it    impos- 
sible  to  discover  the  law  of  their  action.     These  non- 
periodic    variations   are   much    greater  in  amount   and 
more  rapid  in  action    than  any  of  the  others.     For  ex- 
ample, in  a  trial  made  for  the  purpose  of  this  record, 
under    apparently   favorable    circumstances,   the    baro- 
metrically-determined   difference    of    elevation    of  two 
points  having   a   difference   of   elevation   of   about   100 
feet  and  being  25  miles  apart  on  a  plain,  was  in  error 
about  400  per  cent,  owing  mainly  to  non-periodic  grad- 
ient.    Usually  these  variations  are  greater  immediately 
before  and  after  a  storm. 

The  errors  arising  from  non-periodic  gradients  are 
approximately  proportional  to  the  force  of  the  wind 
and  to  the  horizontal  distance  between  the  two  sta- 
tions. 

355.  Permanent  Gradient.      Since   the  atmosphere,  if 
undisturbed,  would  have  no  gradients,  and  since  every 
disturbance  produces  them,  it  is  easy  to  understand  that 
any  continuous    disturbance  will    be    accompanied  by 
permanent  gradients.     The  excess  of  solar  heat  received 
in  the  tropics,  as  compared  with  the  polar  regions,  is  of 
the    nature   of  a  continuous   disturbance,   and   sets   in 
motion  the  great  currents  of  the  atmosphere  and   the 
ocean.     The  joint  action  of  these  causes  gives  rise  to 
a  great    system  of  permanent  gradients.     The  annual 
gradients  are  only  variations  of  the  permanent  gradient, 
caused  by  the  annual  progress  of  the  sun  from  tropic  to 
tropic. 

The  topographic  conditions  of  the  earth's  surface  also 
probably  cause  a  somewhat  permanent  gradient. 

356.  Conclusion  as  to  Gradient  Errors.     The  result  ob- 


ART.  3]  THE    OBSERVATIONS. 


tained  by  barometric  leveling  may  be  in  error  to  almost 
any  degree  owing  to  barometric  gradient  and  to  changes 
in  the  gradient.  A  description  will  presently  be  given 
(see  §§  371-81)  of  several  methods  of  making  the  obser- 
vations which  eliminate  at  least  part  of  the  errors  due 
to  gradient. 

357.  Temperature  of  the  Air.  Variations  in  temperature 
is  the  chief  cause  of  changes  in  barometric  pressure,  but 
the  variation  in  the  temperature  of  the  air  has  another, 
and  generally  a  more  serious,  effect  upon  the  results 
obtained  by  barometric  leveling.  Let  A  and  B,  Fig.  84, 
page  310,  be  two  stations  the  difference  of  elevation  of 
which  is  to  be  obtained  from  observations  of  the 
barometer  and  thermometer  made  at  each.  Assumed 
that  the  pressure  observed  at  B  is  the  same  as  that  at 
C  —  vertically  over  A  and  on  a  level  with  B.  To  use  any 
of  the  barometric  leveling  formulas,  the  temperature 
of  the  column  AC  must  be  known;  and  in  applying 
any  of  these  formulas  it  is  assumed  that  the  mean  tem- 
perature of  this  column  is  equal  to  the  mean  of  the 
temperatures  observed  at  A  and  B. 

How  admissible  this  assumption  is  will  appear  at  once 
when  the  manner  in  which  the  air  acquires  and  loses 
heat  is  recalled.  The  body  of  the  atmosphere  is  heated 
directly  by  the  sun,  and  gives  off  its  heat  by  radiation 
into  space.  The  surface  of  the  earth  is  heated  and 
cooled  in  the  same  manner,  but  many  times  more  rapidly, 
so  that  by  day  it  is  always  much  warmer  than  the  body 
of  the  air,  and  by  night  it  is  much  cooler.  A  layer  of 
air  next  to  the  earth  receives  its  warmth  from  the  earth, 
and  hence  differs  widely  in  temperature  from  the  re- 
mainder of  the  atmosphere.  Not  only  is  the  greater 
part  of  the  column  inaccessible  to  us,  but  that  portion 
to  which  our  observations  are  restricted  is  the  portion 
least  representative  of  all. 

"  By  measuring   the    difference   of   elevation    of  two 


314  fcAROMETERS.  [CHAP,  xii 

points  with  the  spirit  level,  reversing  the  barometer  for- 
mula, and  computing  the  temperature  of  the  air  column, 
it  has  been  found  that  in  middle  latitudes  the  average 
daily  range  of  the  temperature  of  the  body  of  the  air  is 
about  4°,  of  the  superficial  layer  from  10°  to  20°  near 
the  seashore,  and  from  20°  to  35°  in  the  interior  of  con- 
tinents." *  There  is  a  stratum  of  air  near  the  surface  of 
the  earth  which  oscillates  daily  throirgh  this  wide  range, 
while  the  temperature  of  the  upper  and  larger  portion 
of  the  column  AC  is  relatively  constant;  and  therefore 
the  mean  of  the  observed  temperatures  absolutely  fails 
to  give  the  mean  temperature  of  the  column  AC  as  re- 
quired in  the  formula. 

358.  Nor  does  the  trouble  end  here.     Whenever  the 
ground  layer  is  cooler  than  the  air  above,  it  is  of  course 
heavier,  and,  like  any  other  heavy  fluid,  it  flows  down 
hill  and   accumulates  in  valleys,  forming  lakes  of  cold 
air.     The  nightly  layer  of  abnormally  cool  air  is  there- 
fore thinner  on  eminences  than  in  valleys,  and  the  con- 
trast increases  as  the  night  advances.     When  the  condi- 
tions are  reversed  so  that  the    lower  layer  is  warmer 
than  the  air  above  it,  it  has  a  tendency  to  rise,  but  ac- 
complishes the  change  in  an  irregular  manner,  breaking 
through  the  immediately  superior  layer  here  and  there 
and  rising  in  streams  which  spread  out  in  sheets  wher- 
ever the   conditions   of  equilibrium   are   reached.     Ob- 
servers in  balloons,  as   they  ascend   or  descend,  rarely 
find  an  orderly  succession  of  temperatures.     If,  there- 
fore, we  could   in   any  way  determine  the  temperature 
of  some  point  in  the  upper   portion  of  the  column  AC, 
we  should  still  be  unable  to  deduce  the  mean  tempera- 
ture of  the    column    with   any  considerable   degree  of 
accuracy. 

359.  Concision  as  to  Error  in  the   Temperature  of  the 

*  G.  K.  Gilbert,  in  the  Report  of  the  U.  S.  Geological  Survey  for  1880- 
81,  p.  421. 


ART.  3]  THE    OBSERVATIONS.  3lg 

Atmosphere.  For  an  error  of  5°  F.  in  the  mean  tempera- 
ture of  the  level  stratum  of  air  between  the  two  stations, 
the  resulting  error  is  approximately  i  per  cent  of  the 
difference  of  elevation  (see  §  390).  This  error  may  even 
under  favorable  conditions  be  two  or  three  times  this 
amount,  and  under  unfavorable  conditions  five  or  six 
times  as  large. 

Several  methods  of  eliminating  the  error  due  to  the 
uncertainty  in  the  temperature  of  the  air  will  be  de- 
scribed presently  (§§  371-81);  but  even  with  the  utmost 
care  and  the  most  elaborate  system  of  observations,  the 
determination  of  the  temperature  of  the  air  is  the  chief 
source  of  error  in  barometric  leveling. 

360.  Humidity.  The  barometer  is  influenced  by  the 
elastic  force  of  the  invisible  aqueous  vapor  suspended  in 
the  atmosphere,  in  the  same  way  that  it  is  influenced 
by  the  dry  air;  and  hence  the  attempt  has  been  made  to 
introduce  a  term  into  the  barometric  leveling  formula, 
which  shall  take  account  of  the  amount  of  this  watery 
vapor.  The  determination  of  the  humidity  of  the  at- 
mosphere involves  essentially  the  same  difficulties  as 
the  determination  of  its  temperature  (see  §§  357-59). 
The  observations  are  made  in  the  stratum  next  to  the 
earth,  in  which  the  amount  of  moisture  is  the  greatest 
and  the  most  variable.  A  change  of  position  of  a  few 
feet,  or  a  slight  variation  in  the  direction  or  force  of  the 
wind,  will  often  cause  a  very  important  difference  in  the 
amount  of  watery  vapor  present. 

The  variations  in  the  hygrometric  state  are  still 
further  increased  by  vaporization  and  condensation. 
Whenever  a  current  of  air  moves  upward  its  tempera- 
ture is  lowered  by  refraction,  and  a  point  may  be 
reached  where  the  accompanying  vapor  can  no  longer 
exist  as  such,  and  is  condensed  to  cloud  or  even  to  rain 
or  snow.  On  the  other  hand,  whenever  a  current  of 
air  moves  downward  its  capacity  for  moisture  is  in- 


BAROMETERS.  [CHAP.  Xll 


creased,  and  it  acquires  the  power  to  take  up  water 
from  whatever  moist  surface  it  comes  in  contact  with. 
At  the  surface  of  the  earth  there  is  an  almost  continu- 
ous passage  of  moisture  from  ground  to  air,  only  a 
part  of  which  is  returned  as  dew.  The  daily  circulation 
of  the  atmosphere  incited  by  the  heat  of  the  sun  carries 
the  moistened  air  upward,  and  eventually  it  is  con- 
densed and  returned  to  the  earth  in  the  form  of  rain  or 
snow;  but  the  condensation  and  succeeding  precipita- 
tion are  exceedingly  irregular. 

The  irregularities  of  humidity  are  greater  proportion- 
ally than  the  irregularities  of  temperature,  but  the 
error  in  the  difference  of  altitude  due  to  humidity  is 
less  than  that  due  to  temperature,  because  humidity  is 
a  much  smaller  factor  of  the  hypsometric  problem.  In 
a  very  general  sense,  in  temperate  climates  near  the  sea 
level,  the  amount  of  vapor  in  the  atmosphere  varies 
from  0.2  to  0.4  of  an  inch,  or  about  one  hundredth  of  the 
whole. 

361.  Conclusion  as  to  Humidity.     As  the  methods  em- 
ployed to  eliminate  errors  of  gradient  and  temperature 
(§§  37x~8i)  a^so  eliminate  errors   due   to  humidity,  the 
subject  will  not  be  discussed  further  here.     See  §§  393- 

95- 

362.  Instrumental  Errors.     Mercurial  Barometer.     The 
errors  that  may  exist  in  a  mercurial  cistern  barometer 
are  (i)  error  in  the  position  of  the  index,  (2)   imper- 
fect graduation  of  the  scale,   (3)  error  in  the   position 
of  the  zero  and  in  the  graduation  of  the  attached   ther- 
mometer, (4)  impure  mercury,  and  (5)  air  in  the  tube. 
The   first  is  eliminated    by  comparing  the  instrument 
with  a  standard  barometer,  and  either  re-adjusting  the 
zero  of  the  scale  or  noting  the  correction  to  be  applied 
to    the    reading.      With    a    good     barometer    properly 
handled,  the  other  errors  will  be  inappreciable. 

363.  Aneroid  Barometer.     The  index   of    the    aneroid 


ART.  3]  THE    PRACTICE.  31 7 

is  easily  displaced,  and  hence  it  should  be  frequently 
compared  with  a  mercurial  barometer.  The  attached 
thermometer  also  should  be  tested.  See  §§  343-45. 

364.  Errors  of  Observation.    Mercurial  Barometer.    The 
principal  errors  of  observation    are   in   making  contact 
between  the  ivory  point  and  the  mercury,  and  in  adjust- 
ing the  vernier  to  coincide  with  the  top  of  the  mercurial 
column  (see  §  338).      Gilbert,  from  a  comparison  of  three 
hundred  and  sixty   pairs  of  observations   made  by  the 
Signal    Service   and   by  the  Geological  Survey,  found 
the  average  error  of  observation  to   be  a  trifle  less  than 
three  thousandths  of  an  inch.*      This  difference  does 
not    involve    the    personal    equation  between  two    ob- 
servers, which,  even  for  experts,  may  be  nearly  as  much 
more.f 

365.  Aneroid  Barometer.     Owing  to  parallax,  there  is 
a  liability  of  considerable  error  in  reading  the  position 
of  the  index.     The  amount  of  this   error  will   depend 
mainly  upon  the  distance  the  index  is  from  the   scale; 
but  as  the  aneroid  is  not  an   instrument  of  any  consid- 
erable precision,  this  subject  will  not  be  discussed  fur- 
ther.    See  §§  343-45- 

366.  Thermometers.     There  is  liability  of  error  owing 
to  parallax  in  reading  the  attached  and  detached   ther- 
mometers.    With   the   mercurial   barometer  an  error  of 
o.ic7.  in  reading  the  attached   thermometer  produces 
an   error  of  about  0.3  ft.  in  the  elevation.     Notice  that 
this  source  of  error  is  independent  of,  and  in  addition 
to,  that  discussed  in  §§  357-59.     The  latter  is   usually 
very  much  the  greater. 

The  chief  source  of  error  in  determining  the  tempera- 
ture of  the  air  or  of  the  instrument,  is  in  the  exposure 
of  the  thermometer.  It  should  be  very  carefully 
shielded  from  both  the  direct  and  reflected  rays  of  the 

*  U.  S.  Geological  Report,  1880-81,  p.  542. 
t  U.  S.  Coast  Survey  Report,  1870,  p.  79. 


318  BAROMETERS.  [CHAP.  XII 

sun,  and  the  observer  should  be  careful  that  the  heat  of 
his  own  body  does  not  affect  the  thermometer  (§  338). 

The  brass  scale  is  usually  so  thin  that  it  undergoes 
changes  of  temperature  more  rapidly  than  the  mercury; 
and  therefore  if  the  temperature  of  the  surrounding 
air  be  gradually  raised,  the  brass  scale  responds  more 
promptly  than  the  mercurial  column  and  becomes  rela- 
tively too  long,  while  the  reverse  takes  place  if  the 
temperature  is  lowered.  The  result  is  that  a  rising 
temperature  gives  too  low  a  barometric  pressure,  and  a 
falling  temperature  too  high  a  one.  If  the  change  of 
temperature  is  rapid,  the  error  from  this  cause  may 
amount  to  ten  or  fifteen  thousandths  of  an  inch,  which 
corresponds  to  a  difference  of  elevation  of  10  or  15  feet. 
The  precaution  generally  recommended  is  to  put  the 
barometer  in  position  and  leave  it  with  unchanged  con- 
ditions for  fifteen  or  twenty  minutes  before  making  the 
observation. 

367.  Wind.  The  wind  may  cause  either  a  condensa- 
tion or  a  rarefaction  of  the  air  in  the  room  in  which 
the  barometer  is  hung,  or  even  in  the  cistern  of  the 
barometer  itself.  This  effect  will  vary  with  the  velocity 
of  the  wind,  the  position  of  the  openings  with  reference 
to  the  direction  of  the  wind,  the  size  of  the  room,  etc. 
"  On  Mount  Washington,  a  wind  of  50  miles  per  hour 
caused  the  barometer  to  read  0.13  of  an  inch  too  low. 
The  effect  of  the  wind  will  vary  as  the  square  of  its 
velocity.  It  may  be  nearly,  if  not  wholly,  eliminated  by 
having  two  apertures,  one  each  on  the  windward  and 
leeward  side  of  the  enclosed  space."  * 

The  wind  has  a  similar  effect  when  the  instrument 
is  read  in  the  immediate  vicinity  of  any  body  which 
obstructs  the  wind.  For  example,  if  the  barometer 
is  observed  on  the  windward  side  of  a  mountain,  the 


*G.  K.  Gilbert  in  Report  U.  S.  Geological  Survey  for  1880-81,  p.  562. 


ART.  3]  THE    PRACTICE.  319 

reading  will  be  too  high;  and  if  on  the  leeward,  too 
low.  The  only  wray  to  eliminate  this  error  is  to  select 
stations  not  thus  affected, — although  it  is  not  always 
possible  to  do  so. 

368.  LIMITS  OF  PRECISION.  It  is  sometimes  stated  that 
"the  barometer  is  the  most  accurate  instrument  for 
determining  differences  of  level;"  but  it  needs  only  a 
moment's  reflection  to  see  that  this  can  not  be  true. 
The  following  results,  given  by  Professor  Guyot,*  are 
frequently  quoted  as  showing  the  great  accuracy  of 
barometric  leveling: 

Mount  Blanc,  by  barometer,  15,781  feet, 

by  spirit  level,  15,780  feet. 
Mount  Washington,  by  barometer,  6,291.7  feet, 

by  spirit  level,  6,293  feet. 
In  North  Carolina,  by  barometer,  5,248  feet, 

by  spirit  level,  5,246  feet. 
In  North  Carolina,  by  barometer,  6,701  feet, 

by  spirit  level,  6,711  feet. 

A  few  examples  showing  a  small  error  prove  nothing 
as  to  the  accuracy  of  the  method,  for  the  agreement 
may  be  wholly  accidental.  A  fair  inference  from  the 
remarks  of  Professor  Guyot  accompanying  the  above 
data,  is  that  the  mean  of  several  observations  taken 
during  one  or  two  days  will  generally  give  as  accurate 
results  as  those  above;  but  it  is  probable  that  he  did 
not  intend  to  convey  such  an  impression,  for  the  very 
observations  on  which  he  relies  for  his  spirit  level  alti- 
tude of  Mount  Washington  resulted  in  a  suggestion  to 
modify  the  constants  in  the  barometric  formula. J  In 
one  instance  his  barometric  results  were  in  error  125 
feet  in  a  difference  of  altitude  of  1,780  feet;  and  in 
another  75  feet  in  a  difference  of  altitude  of  6,280 
feet. 


*  Smithsonian  Miscellaneous  Collection,  Vol.  I,  Art.  II,  Group  Dv  p. 
JUS.  Coast  Survey  Report,  1854,  p.  100*. 


320  BAROMETERS.  [CHAP.  XII 

369.  The  difference  of  altitude  computed  from  one, 
or  even  several,  day's  observations  can  not  be  relied 
upon  as  being  more  than  a  rough  approximation.  This 
has  been  shown  by  Williamson,*  who  has  computed 
the  difference  of  altitude  between  Geneva  and  St.  Ber- 
nard by  the  formula  used  in  the  last  three  examples 
quoted  above,  for  every  day  for  two  years,  using  the 
daily  means  of  simultaneous  bi-hourly  observations. 
In  several  cases,  the  difference  between  the  result  by 
the  barometer  and  the  spirit  level  was  more  than  3 
per  cent.  Under  less  favorable  circumstances  the 
errors  were  even  more  than  twice  as  great. f  The  alti- 
tude computed  by  the  monthly  means  of  bi-hourly  ob- 
servations for  different  months  of  the  same  year,  and 
also  for  the  same  month  of  different  years,  differ  as 
much  as  i  per  cent.J 

The  following  differences  between  the  results  by  the 
barometer  and  the  spirit  level  do  not  indicate  that  high 
degree  of  accuracy  in  barometric  hypsometry,  even  when 
a  long  series  of  observations  is  used,  which  was  formerly 
supposed  to  be  attainable  by  this  means.  The  results 
by  the  barometer  were  obtained  by  computing  the  dif- 
ference of  altitude  from  monthly  means  of  the  mean  of 
the  daily  observations,  and  taking  the  mean  for  the 
time  stated. § 

Vera  Cruz  and  City  of  Mexico,  I  year's  observations, 

+  5  metres  in  2,282  metres. 
Sacramento  and  Summit,  3  years'  observations, 

—  24  feet  in  6,989  feet. 
Portland  and  Mt.  Washington,  6  years'  observations, 

+  37  feet  in  6,289  feet- 
Geneva  and  St.  Bernard,  12  years'  observations, 

—  2.6  metres  in.  2,070  metres. 

*  Williamson's  On  the  Barometer,  pp.  194-205. 

t  U.  S.  Geological  Survey  Report  for  1880-81,  pp.  456-59. 

\  Williamson's  On  the  Barometer,  p.  236. 

S  U.  S.  Coast  and  Geodetic  Survey  for  1881,  p.  254. 


OF  THK 

UNIVERSITY 


_. 

ART.  3]  THE    PRACTIClL  12  1 


For  an  interesting  comparison  of  the  absolute,  and 
also  the  relative,  errors  of  the  various  methods,  see 
Chap.  Ill  of  the  Report  of  U.  S.  Geological  Survey  for 
1880-81.  For  additional  data  concerning  the  accuracy 
of  barometric  leveling,  see  Report  of  the  U.  S.  Coast 
and  Geodetic  Survey  for  1870,  p.  88;  ibid.,  1871,  pp. 
154-75;  ibid.,  1876,  pp.  356-76;  Williamson's  On  the 
Barometer,  p.  205. 

370.  Although  the  barometer  can  not  be  regarded  as 
a  hypsometric   instrument   of  great  precision,  yet  with 
care  it  can  be  made  to  give  results  with  sufficient  accu- 
racy for  reconnoissance  or  exploration.     For  this  pur- 
pose, it  is  unexcelled  by  any  other  instrument;  and  this 
is  about   the  only  use  the  engineering  profession  can 
make  of  it. 

371.  METHODS  OF  OBSEKVING.    In  §§  348-61  it  was 

shown  that  barometric  leveling  was  liable  to  large 
errors  owing  to  barometric  gradient,  and  to  errors  in 
determining  the  temperature  and  humidity  of  the  at- 
mosphere. In  the  best  work,  the  observations  are  made 
in  such  a  way  as  to  eliminate  part  of  these  errors.  The 
several  methods  employed  to  accomplish  this  will  now 
be  considered. 

372.  Single  Observations.     The  simplest,  and  also  the 
most  common,  method  consists  in    using  only  one  ba- 
rometer, which  is   carried  from   station  to  station.     By 
taking  one  or    more   observations   at  each   station,  the 
errors  of  observation  (§§364-67)  are  nearly  eliminated; 
but  the  vastly  greater  errors  due  to  gradient,  tempera- 
ture, and  humidity  (§§  349-61)  are  undiminished.     Re- 
sults   obtained    by   a   single    observation,    or   even   by 
several  in  quick  succession,  are  only  the  rudest  kinds  of 
approximations  (§354),  and  the  greater  the  horizontal 
and  vertical    distances    between    the    two  stations   the 
greater  the  error. 

Distant   stations  are  sometimes  connected  by  inter- 


322  BAROMETERS.  [CHAP.  XII 

mediate  ones  to  eliminate  changes  of  gradient,  tem- 
perature, etc.,  during  the  time  the  exigencies  of  travel 
require  the  barometer  to  remain  at  the  intermediate 
station.  For  example,  to  determine  the  difference  of 
elevation  between  A  and  C,  make  an  observation  at  A 
and  proceed  towards  C,  making  an  observation  at  B,  an 
intermediate  point,  on  arrival  at  that  station  and  a 
second  one  on  leaving  it;  and  finally  make  an  observa- 
tion at  C.  The  difference  of  elevation  of  A  and  B  is 
determined  from  the  first  two  readings,  and  that  of  B 
and  C  from  the  second  two;  and  the  difference  of  eleva- 
tion of  A  and  C  is  equal  to  the  sum  of  the  partial  differ- 
ences. 

373.  Simultaneous   Observations.     The    errors   due    to 
change  of.  gradient  (§§349-56)  are  partially  eliminated  by 
making  simultaneous  observations  at  the  two  stations. 
If  the  phase  and  the  amplitude  of  the  variation  were  the 
same  at  both  stations,  which  probably  seldom  or  never 
occurs,   simultaneous  observations  would   give    results 
independent  of  changes  in  the  gradient. 

Errors  due  to  gradient  are  still  further  reduced  by 
making  a  number  of  simultaneous  observations  and 
using  the  mean.  This  may  eliminate  the  variable  ele- 
ment, but  fails  to  take  account  of  permanent  gradient 
(see  §  369). 

374.  Observations  at  Selected  Times.     It  is  often  recom- 
mended that  the  observations  be  made  at  certain  hours 
of  the  day,  at  which  time  the  diurnal  gradient  is  supposed 
to  be  zero.     These  times  can  be  determined  only  by  ob- 
servation, and  will   vary  greatly  with   the   state   of   the 
atmosphere,  the  season,  the  locality,  the  elevation,  etc. 
The    U.  S.    Coast    Survey    recommend    the    following 
times.*     They   were   probably  deduced   for  the   middle 
Atlantic   coast.     The  hours  refer  to   the   middle  of  the 

*  Report  for  1876,  p.  349. 


ART.  3]  THE    PRACTICE.  323 

month,  and  other  times  are  to  be  determined  by  inter- 
polation. 

January I         P.M. 

February 10        A.M.  and  4  " 

March 8  "     6 

April 7  30     "        "     7 

May 7  "     7 

June 6.30     "        "     9.30      " 

July 6.30     "        "     9.30      " 

August 7          "        "     7.30      " 

September 8          "        "6 

October 10          "        "     3.30      " 

November 10.30    "         "     2,30      " 

December at  no  time. 

Obviously,  a  single  observation,  even  if  taken  in  ac- 
cordance with  a  table  similar  to  the  above,  could  be 
considered  only  a  rough  approximation  owing  to  the 
liability  of  error  due  to  non-periodic  gradient  (§  354). 
For  a  long  series  of  observations,  the  results  would 
probably  be  more  accurate  if  the  observations  were 
made  in  accordance  with  some  such  scheme;  but  the 
determination  of  the  best  times  at  which  to  make  the 
observations  would  necessitate  an  extensive  prelimi- 
nary series  of  observations  at  short  intervals.  Such  a 
method  is  clearly  inapplicable,  since  the  time  and  expense 
required  to  obtain  approximate  results  by  the  barometer 
would  be  nearly  or  quite  as  great  as  that  required  to 
determine  an  equal  number  of  more  accurate  results  by 
the  stadia  (Chapter  X)  or  by  spirit  leveling  (Chapter 
XI). 

375.  Notice    that    none    of    the    preceding   methods 
eliminate  the  error  due  to  the  fact  that  the  mean  of  the 
observed    temperatures    does    not    represent    the  mean 
temperature  of  the  air  column  (§§  357-59). 

376.  Williamson's  Method.*     This  method  is  specially 
adapted  to  reconnoissances  and  topographical  surveys. 

*  Williamson's  On  the  Use  of  the  Barometer,  pp.  39-42,  150-59. 


324  BAROMETERS.  [CHAP.   XII 

A  centrally  located  station,  called  a  base  station,  is 
chosen,  at  which  the  barometer  and  thermometers  are 
read  at  stated  hours  each  day  for  several  days.  In  the 
meantime,  itinerary  observers  make  observations  at  the 
points  whose  elevations  are  to  be  determined,  taking 
pains  to  have  each  observation  correspond  in  time  with 
one  of  the  observations  at  the  base  station.  In  the 
progress  of  the  itinerary  survey,  a  series  of  observa- 
tions, similar  to  those  at  the  base  station,  are  made  as 
frequently  as  practicable  at  semi-permanent  camps. 
The  object  of  the  series  at  the  base  station  and  at  the 
semi-permanent  camps  is  to  ascertain  the  nature  of  the 
diurnal  variation  of  pressure  and  temperature. 

The  barometric  readings  at  the  base  stations,  after 
being  corrected  for  temperature  of  the  instruments 
(§  389)'  are  plotted  upon  ruled  paper  so  as  to  exhibit 
their  curve,  and  all  readings  shown  by  inspection  to 
be  influenced  by  abrupt  and  violent  atmospheric  dis- 
turbances, such  as  thunder  storms,  are  discarded,  their 
places  being  filled  by  interpolated  values.  From  the 
corrected  observations  at  the  base  stations,  a  correction 
is  deduced,  which,  being  applied  to  the  several  baro- 
metric readings,  reduces  them  to  the  daily  mean.  Ap- 
plying this  correction  eliminates  at  least  part  of  the 
effect  of  the  diurnal  gradient  (§352). 

Instead  of  determining  the  temperature  of  the  air 
column  from  the  temperature  at  the  time  of  observing, 
the  mean  temperature  of  the  day  is  used.  This  can  be 
quite  accurately  determined  at  the  base  stations,  but  is 
only  approximately  known  at  the  other  stations.  No- 
tice, however,  that  the  mean  of  the  observed  tempera- 
tures will  not  be  the  mean  temperature  of  the  stratum 
of  air  included  between  the  two  stations  (§§  357-59). 

The  difference  of  altitude  can  be  computed  from  the 
reduced  barometric  readings  and  the  mean  daily  tem- 
perature, by  using  any  statical  formula  (Art.  4).  Wil- 


ARTI  3]  THE    PRACTICE.  325 

liamson  himself   used  his   translation  of  Plantamour's 
formula  (§  401). 

377.  Whitney's  Method.     From  observations  made  in 
connection  with  the  Geological  Survey  of  California,  a 
series   of    corrections  were    deduced    for    reducing    the 
barometric  readings  made  at  different  hours  of  the  day 
of   the   different  days  of   the  different   months,  and  for 
different  altitudes,  to  the  daily  mean  for  the  year. 

These  corrections  can  only  be  used  in  the  neighbor- 
hood in  which  the  observations  on  which  they  were 
based  were  made.  Similar  tables  made  for  different 
climates  differ  materially  from  each  other.  For  tables 
constructed  upon  this  principle  for  the  climates  of  Ger- 
many, Philadelphia,  and  Greenwich,  respectively,  see 
Smithsonian  Miscellaneous  Collection,  Vol.  I  (3d  Edi- 
tion), Art.  Ill,  Group  D,  pp.  80-82,  86,  and  93-94. 

378.  Plantamour's  Method.    In  the  hypsometric  survey 
of  Switzerland,  Plantamour  made  simultaneous  obser- 
vations of  the  barometer,  thermometer,  and  psychrom- 
eter  at  Geneva,  St.  Bernard,  and   at  the  station  whose 
height  was  to  be  determined.     The  approximate  differ- 
ence of  altitude  between  the  new  station  and  Geneva, 
and    between    it   and    St.   Bernard,    and    also    between 
Geneva    and    St.    Bernard,   were   computed    by    Plan- 
tamour's formula  (§  401).     The  computed  difference  of 
elevation  between  Geneva  and  St.   Bernard  compared 
with   the  difference  of  altitude  determined  by  the  spirit 
level,  gave  a  correction  to  be  applied  to  the  computed 
difference  for  the  new  station.     The  ingenious  details 
of  the   computation   are  too   complex   to  be  described 
here. 

Marshall  employed,  in  the  geographical  surveys  in 
the  Rocky  Mountains,  a  method  somewhat  similar  to 
the  above.* 

*  U.  S.  Geological  Surveys  West  of  the  zooth  Meridian,  Vol.  II,  pp.  522, 
523. 


326  BAROMETERS.  [CHAP.  XII 

379.  Riihlmann's  Method.     This   method    differs  from 
Plantamour's  (§  378)  chiefly  in  that  Riihlmann   makes 
use  of  the  pressure  and  the  moisture  factors  observed 
at  two  stations  whose  difference  in  level  is  known,  to 
compute  the  temperature  of  the  intervening  air  column. 
The  temperature  thus  derived  he  afterwards  applies  in 
the  computation  of  the  unknown  difference  in  level  of 
two  other  stations. 

This  method  is  applicable  only  to  a  detailed  topo- 
graphical survey,  and  requires  such  an  elaborate  series 
of  observations  at  base  stations  that  the  time  and  ex- 
pense are  nearly,  or  quite,  as  great  as  that  required  to 
determine  the  results  with  the  stadia  (Chap.  X)  or  the 
spirit  level  (Chap.  XI). 

380.  Gilbert's  Method.*     This  method  is  practically  a 
combination    of     Plantamour's    and    Riihlmann's,    and 
requires  simultaneous  observations  of  the  barometer  at 
three   stations,   the  vertical    distance    between    two    of 
which  is  known,  but  does  not  require  observations  of 
the  temperature  and  humidity.     From  the  known  dif- 
ference between  two  of  the  stations  and  the  observations 
at  each,  the  actual  density  of  the  air,  which  is  dependent 
upon  the  pressure,  temperature,  and   humidity,  may  be 
determined  by  reversing  the  ordinary  barometric  for- 
mula.    The  true  density  is  then  used  to  compute  the 
difference  of  elevation  between  either  of  these  stations 
and    the    third.     For    the    formula    employed    in     this 
method,  see  §  407. 

This  method  has  not  proved  as  satisfactory  in  prac- 
tice as  was  hoped  when  it  was  proposed.  It  is  simply 
another  attempt  to  secure  accurate  results  by  a  modifi- 
cation of  a  general  method  which  is  inaccurate  in  its 
essential  features. 

381.  Conclusion    as    to    Methods.     Unfortunately    all 
methods    of    eliminating    gradient    and     temperature 

*  Report  oL  U.  S.  Geological  Survey  *or  ^080-81,  pp.  ^05-561. 


ART.  4]  BAROMETRIC    FORMULAS.  327 

errors  involve  considerable  time  and  expense,  and  even 
then  do  not  thoroughly  accomplish  the  desired  end, 
which  shows  that  when  great  accuracy  is  desired  the 
barometer  should  be  dispensed  with  altogether,  and  the 
difference  of  elevation  be  determined  by  some  other 
means. 

ART.  4.     BAROMETRIC  FORMULAS. 

382.  Barometric   leveling  formulas   may   be   divided 
into  two   classes;    viz.,  (A)    those   that   assume   the   at- 
mosphere   to    be  in   statical  equilibrium,  and  (B)  those 
that  take  account  of  the  fact  that  the   air   is   more  or 
less   in   motion.     The   first  will    be   designated   statical 
formulas,  and  the  latter  dynamical.     The  former  are  by 
far  much  more  frequently  used. 

A.  Statical  Formulas. 

383.  ASSUMPTIONS.  All  the  barometric  leveling  formu- 
las in  common  use  are  dependent  upon  the  assumption 
that  the  air  is  in  a  state  of  statical    equilibrium.*     If 
this  state  were  possible,  we  might  suppose  the  whole 
atmosphere  to  be  arranged  in  a  system  of  horizontal 
layers,  each  of  which  is  denser  than  the  one  above  it 
and   rarer  than   the  one  below  it,  each  being  uniform 
throughout  in  temperature  and  humidity. 

But  the  air  is  never  in  a  state  of  statical  equilibrium, 
but  is  perpetually  undergoing  local  changes  of  pressure, 
temperature,  and  humidity.  For  example,  during  the 
day  the  sun  imparts  a  certain  amount  of  heat  to  the 
whole  atmosphere,  but  a  much  higher  temperature  is 
given  to  the  ground  and  by  it  communicated  to  the 
contiguous  layer  of  air.  At  night  the  atmosphere  loses 
heat  by  radiation  to  space,  but  the  ground  loses  it  more 

*  There  are  only  three  not  thus  included  (see  §§  406-8),  and  practically  they 
are  not  used. 


328  BAROMETERS.  [CHAP.   Xll 

rapidly  and  imparts  its  low  temperature  to  the  lowest 
stratum  of  air.  The  lower  strata,  therefore,  have 
exceptional  warmth  by  day  and  exceptional  coolness  by 
night.  Again,  if  the  air  is  moist,  during  the  day  it 
intercepts  a  greater  quantity  of  solar  heat  than  if  it 
were  dry,  so  that  a  less  quantity  reaches  the  ground; 
but  at  night  the  moisture  checks  radiation  from  the 
ground.  The  power  of  the  earth's  surface  to  receive  or 
part  with  heat  varies  with  its  character;  naked  rocks 
and  cultivated  fields,  bare  earth  and  grass,  forest  and 
snow  are  affected  very  differently  by  the  heat  rays  of 
the  sun,  and  exert  equally  diverse  influences  on  the 
adjacent  air,  so  that  one  tract  of  land  is  often  in  a  con- 
dition to  heat  the  air  while  an  adjacent  tract  is  cooling 
it.  Then,  too,  the  sun's  heat  is  unequally  distributed 
through  the  year;  outside  the  tropics  there  is  a  pro- 
gressive accumulation  of  heat  through  summer  and  a 
progressive  loss  through  winter.  The  ocean  undergoes 
less  change  of  temperature  than  the  land,  and  its  rate 
of  change  is  slower,  so  that  there  is  frequent,  and 
indeed  almost  continuous,  contrast  of  condition  be- 
tween it  and  the  contiguous  land.  As  a  result  of  all 
these  influences,  together  with  others  that  might  be 
enumerated,  the  equilibrium  of  the  air  is  constantly 
disturbed,  and  the  winds,  which  tend  to  readjust  it, 
are  set  in  motion. 

The  temperature  of  the  air  is  continually  modified 
by  external  influences;  the  static  order  of  densities  is 
broken  and  currents  are  set  in  motion;  and  the  circula- 
tion and  the  inequalities  of  temperature  also  conspire 
to  produce  inequalities  of  moisture.  Every  element  of 
equilibrium  is  thus  set  aside,  and  the  air  is  rendered 
heterogeneous  in  density,  temperature,  and  humidity. 

384.  All  of  the  common  barometric  formulas  are 
dependent  upon  two  assumptions;  viz.,  (i)  that  a  dif- 
ference of  pressure  is  due  only  to  a  difference  of  eleva- 


ART.  4]  STATICAL    FORMULAS.  $2$ 

tion  ;  and  (2)  that  the  mean  of  the  temperatures  of 
the  air  at  the  two  stations  represents  the  mean  tem- 
perature of  the  level  stratum  of  air  between  them.  A 
few  of  the  formulas  involve  also  a  third  assumption; 
viz.,  that  the  mean  humility  of  the  air  at  the  two  sta- 
tions is  the  same  as  the  humidity  of  the  layer  between 
them. 

A  consideration  of  the  facts  stated  in  §  383  and  dis- 
cussed more  fully  in  §§  348-61,  will  show  that  any 
formula  dependent  upon  the  preceding  assumptions 
can  be,  at  best,  only  approximate.  In  §§  371-81  were 
explained  several  methods  of  reducing  the  errors  due 
to  the  above  assumption;  but  notice  that  errors  of 
gradient,  temperature,  and  humidity  were 
eliminated  by  the  methods  of  making  the 
observations  and  computing  the  results, 
rather  than  by  the  use  of  any  particular 
formula.  -C 

385.  FUNDAMENTAL  RELATIONS.    Let  A  and 

B,  Fig.  85,  represent    the  two    stations,  and 
assume  that  it  is  required  to  determine  the 
vertical    distance  between    them.      A  and  B 
are   not    necessarily,    and    not    even    usually? 
in    the   same  vertical    line.     Let  C  represent      FlG-8s- 
any  point  in  AB,  and  D  a  point  an  infinitesimal  dis- 
tance below  C. 

Let  a0  =  the  weight  of  a  cubic  inch  of  dry  air  at  the 
sea  level,  in  latitude  45°,  at  32P  F.,  when 
the  barometer  stands  at  29.92  inches. 
a  —  the  weight  of  a  cubic  inch  of  air  under  the 
pressure,  temperature,  etc.,  existing  be- 
tween C  and  D. 

c  =  the  co-efficient  of  expansion  of  air. 
jy0  =  the  height  of  the  barometric  column  at  the 
sea  level  in  latitude  45°,  —  29.92  inches. 


330  BAROMETERS.  [CHAP.  XII 

If  =  the  height,  in  inches,  of  the  barometer  at  the 

upper  point,  reduced  to  32°  F. 
H^  =  the  height,  in  inches,  of  the  barometer  at  the 

lower  point,  reduced  to  32°  F. 

MO  =  the  weight  of  a   cubic  inch   of  mercury  at 
the  sea  level,  in  latitude  45°,  when  the  ba- 
rometer reads  29.92  inches. 
m'  =  the  weight  of  a  unit  volume  of  mercury  at 

the  upper  station. 
ml  =  the  weight  of  a  unit  volume  of  mercury  at 

the  lower  station. 

0  =  the  mean  latitude  of  the  two  stations. 
P  —  the  pressure  per  square  inch,  at  D. 
dP  =  the  difference  in  pressure  between  C  and  Z>, 

/>.,  dP  is  the  differential  pressure. 
P9  =  the   pressure   per  square  inch,   at    the    sea 

level,  in  latitude  45°. 
P'  =  the  pressure  at  the  upper  station. 
Pl  =  the  pressure  at  the  lower  station. 
R  =  the  radius  of  the  earth. 
T—  the  mean   temperature  of  the  layer  of  air 

between  A  and  B. 
Tf  =  the   temperature   of    the   air  at   the   upper 

station. 
Tt  =  the   temperature   of    the   air  at   the   lower 

station. 

X  =  the  difference  in  elevation. 
Z  —  the  elevation  of  the  lower  station  above  the 

sea  level. 

386.  It  is  clear  that  the  increase  of  pressure  from  C 
to  D  is  equal  to  the  weight  of  a  column  of  air  having 
a  unit  section  and  the  height  CD.  Expressing  this  in 
the  language  of  the  calculus,  we  have 

—  dP  —  a  dx (i) 


ART.  4]  STATICAL   FORMULAS.  33! 


By  Marriotte's  law,  a  :  a0  :  :  P  :  P0  ,  from  which 


(2) 


Equation  (2)  gives  the  weight  of  a  unit  of  air  at  32 
F.,  and  for  any  other  temperature  T9 


Equation  (3)  gives  the  weight  of  a  unit  of  air  at  lati- 
tude 45°,  and  for  any  latitude  0, 

P_  I  _  I  _  v 

**/>       1+CT      I  +0.002,60  COS  2   0* 

Equation  (4)  gives  the  value  of  a  at  the  sea  level,  and 
for  any  other  altitude  (Z  -\-  X), 


~~a 


po    IJrcT    i  +  0.002,60  cos  2(p 


Combining    equations    (i)    and    (5),   integrating,   and 
transposing, 

P0  P1        *— 

TT~    ™'*®&~D'  ~    i    T^r 
dn  f      I  H-  6  2 


I  -j-  O.OO2,6o  COS  20  ^2 

A  =  -#owo  =  29-92  W0.       .       .       .  (7) 

-  i          *~-i     i  /o\ 

"D7    ~      rr/T  , Vb; 


332 


BAROMETERS. 


[CHAP,  xii 


The  weight  of  a  body  varies  inversely  as  the  square 
of  the  distance  from  the  center  of  the  earth,  and  there- 
fore 


m 


(R  +  Z)a 


(9) 


Combining  equations  (6),  (7),  (8),  and  (9),  we  get 


- 


(l-}-0.002,6oCOS  20) 


(10) 


The  quantities  at  the  right  of  the  brace  are  three 
factors  of  the  second  member  of  the  equation. 

Substituting,  in  equation  (10),  the  sum  of  the  loga- 
rithms for  the  logarithm  of  the  product,  passing  to 
common  logarithms,  and  expressing  Xy  Z,  and  -#,  in  feet ', 


=  5-744.- log 


(l  -(-  0.002,60  COS  20) 

X 


2l°g 


i  + 


Notice  that  the  preceding  equation  involves  no  ap- 
proximations. 

387.  Equation  (n)  includes  the  principal  relations 
involved  in  determining  differences  of  height  with  the 
barometer.  The  formula  to  be  used  in  practice  has 


ART.  4]  STATICAL    FORMULAS.  333 

been  given  differently  by  different  investigators,  accord- 
ing to  the  values  chosen  for  the  constants,  to  the  indi- 
vidual preference  for  one  form  over  another,  and  to  the 
degree  of  refinement  desired. 
388.  The    Constant.     The   value    of    the    term    5.744 

...— -°,  generally  known  as  the  barometric  co-efficient, 
will  depend  upon  the  values  for  the  densities  of  air  and 
mercury  which  are  used.  Boit  and  Arago  found  — - 

0 

=  10,467,*  which  makes  the  barometer  co-efficient 
60,096.3  ft.  (18,317  meters).  Regnault's  values,*  which 
are  the  most  recent  and  probably  the  most  accurate, 
give  60,384  feet  (18,404.8  meters). 

Raymond  in  1803  found  *  the  value  of  the  barometric 
co-efficient  by  determining  the  value  it  should  have  to 
make  the  results  by  the  formula  agree  with  those  fur- 
nished by  trigonometrical  leveling.  The  value  ob- 
tained in  this  way  is  60,158.6  ft.  (18,336  meters).  Even 
under  the  most  favorable  circumstances,  the  observa- 
tions, eight  in  all,f  were  too  few  to  determine  such  a 
co-efficient  with  sufficient  accuracy,  owing  to  the  varia- 
tions in  the  barometric  pressure  (§§  349-56),  the  tem- 
perature (§§  357-59),  the  humidity  (§  360),  and  the 
refraction  (§  318).  The  method  employed  by  Ray- 
mond is  the  least  accurate  of  any,  although  his  co-effi- 
cient is  more  frequently  used  than  any  of  the  others. 

One  of  the  author's  students  determined  the  baro- 
metric co-efficient  by  reversing  Laplace's  formula, — 
equation  (13),  page  339, — and  using  the  barometric 
readings  made  by  the  U.  S.  Weather  Bureau,  together 
with  the  difference  of  level  of  the  stations.  The 
results  are  not  very  satisfactory  owing  to  the  uncer- 

*  Smithsonian    Miscellaneous  Collections,  Vol.  I,  Guyot's   Meteorological 
Tables,  Group  D,  page  9. 
f  Report  U.  S.  Coast  and  Geodetic  Survey,  1881,  p.  235. 


334  BAROMETERS.  [cHAP.  XTt 

tainty  in  the  values  for  the  differences  of  level.  The 
co-efficient  obtained  from  twelve  pairs  of  stations, 
using  the  mean  of  tri-daily  observations  for  three  to 
twelve  years,  is  60,156  feet.* 

For  a  discussion  showing  that  the  errors  due  to 
gradient  and  temperature  are  very  much  greater  than 
the  errors  due  to  the  differences  between  the  several 
barometric  constants,  see  Williamson's  On  the  Use  of 
the  Barometer,  pages  221-33. 

389.  Temperature  of  the  Barometer. — Before  insert- 
ing the  barometric  readings  in  equation  (u),  page  332, 
they  must  be  reduced  to  the  corresponding  heights 
they  would  have  at  a  common  temperature.  This  cor- 
rection may  be  made  in  either  of  two  ways;  viz.,  (i)  one 
barometer  may  be*  reduced  to  the  temperature  of  the 
other,  or  (2)  both  may  be  reduced  to  any  other  temper- 
ature assumed  as  a  standard. 

i.  The  co-efficient  of  expansion  of  mercury  is  o.ooo,- 
100,1  for  i°  F.;  and  that  of  brass,  of  which  the  scales 
are  generally  made,  is  0.000,010,4.  The  difference — the 
relative  co-efficient  of  expansion  of  mercury — is  o.ooo,- 
089,7.  For  the  centigrade  scale  this  difference  is 
0.000,161,41.  If  h'  represents  the  height  of  the  barom- 
eter at  the  upper  station  reduced  to  the  temperature 
of  the  lower,  and  f  and  /,  the  temperature  of  the 
barometers  at  the  upper  and  lower  stations  respectively, 

h'  =  ff'[l   -  d(f  -  /,)] (12) 

in  which  d  stands  for  one  of  the  above  differences, 
according  to  the  kind  of  thermometer  used.  When 
this  method  is  employed,  a  term  is  inserted  in  the 
formula  to  correct  for  the  difference  in  temperature  of 
the  barometers.  For  an  example  see  §  400. 

*  Bachelor's  Thesis  of   Edward   E.  Ellison,  Class  of  '88,    University  of 
Illinois. 


ART.  4]  STATICAL    FORMULAS.  335 

2.  To  reduce  both  readings  to  a  common  tempera- 
ture, as  for  example  the  freezing  point  of  water,  apply 
equation  (12)  to  both  readings,  considering  tl  to  repre- 
sent the  temperature  of  melting  ice  (32°  F.  or  o°  C.) 
and  f  the  reading  of  the  attached  thermometer.  Nu- 
merous tables  have  been  computed  for  facilitating  this 
reduction;  for  example,  Smithsonian  Miscellaneous 
Collection,  Vol.  I,  Guyot's  Collections  (3d  ed.),  Group 
C,  pp.  61-127;  ibid.,  Group  D,  pp.  30,  46,  53,  etc.;  Lee's 
Tables  (3d  ed.),  pp.  152-59;  Williamson's  On  the  Use  of 
the  Barometer,  pp.  1-64  of  the  appendix. 

390.  Temperature  of  the  Air.     The  term   (i  -f  c  T)   is 
frequently  called  the  temperature  term.     The  name  is 
not  fortunate,  since  some  barometric  leveling  formulas 
have  also  a  term  to  correct  for  the  difference  of  tem- 
perature of  the  barometers  (§  389). 

The  co-efficient  c  is  equal  to  0.003,75  Per  degree  cen- 
tigrade. It  is  usually  approximated  at  0.004,  which 
makes  some  allowance  for  the  greater  expansive  power 
of  the  watery  vapor  contained  in  the  atmosphere. 

The  mean  temperature  of  the  layer  of  air  between 
the  two  stations  is  usually  assumed  to  be  equal  to  the 
mean  of  its  temperature  at  the  two  stations.  Under 
this  assumption,  which  however  may  be  greatly  in 
error  (see  §§  357-58)>  ?=  i(^  +  ^')>  and  the  temper- 
ature term  becomes 

2(2*  -4-  T) 

j  _| — l H  for  centigrade  degrees, 

and 

rrif       I         fji    /- 

i  -| for  Fahrenheit  degrees. 

900 

391.  Latitude  Term.     The  term  (i  -f-  0.002,60  cos  20) 
of  equation   (IT),   page   332,    is  known   as  the    latitude 


536  BAROMETERS.  [CHAP.   XII 

term.  A  few  formul  as  still  contain  an  older  and  less 
accurate  co-efficient  of  cos  20  than  the  above.  How- 
ever, this  correction  is  unimportant,  for  even  at  the 
equator  or  the  poles,  where  this  term  is  a  maximum,  it 
is  only  0.002,60  of  the  difference  of  elevation.  Further- 
more, in  comparison  with  the  possible  errors  due  to 
gradient,  temperature,  and  observation  (see  §§  349-67), 
the  error  caused  by  the  omission  of  the  latitude  term 
is  wholly  inappreciable. 

392.  Altitude  Term.     The  last  term  of  equation  (n), 
page    332,   takes  account  of   the   variation    of   gravity 
with  the  altitude.    This  term  is  always  relatively  small, 
and  may  be  omitted  without   materially  affecting  the 
result.     Furthermore,  owing  to  the  matters  considered 
in  §§  349-67,  as  well  as  the  fact  that  the  barometer  at 
best    can    be    read    only   to    a   thousandth    of   an    inch 
(which  corresponds  to   i  foot  of  altitude),  the  appear- 
ance of  extreme  accuracy  by  retaining  the  altitude  and 
latitude  terms  can  be  regarded  only  as  a  mathematical 
illusion,  inapplicable  to  ordinary  practice. 

393.  Humidity  Term.    In  deducing  equation  (n),  page 
332,  it   was   assumed    that    the   atmosphere    was    com- 
posed exclusively  of  dry  air;  but  it  is  really  a  mixture 
of  air  (oxygen  and  nitrogen),  watery  vapor,  and  carbonic 
acid.     The  carbonic  acid  is  very  small  and  nearly  con- 
stant, and  hence  need  not  be  considered  here  ;  but  the 
watery  vapor  is  both  large  and  variable.      If  dry  air  and 
aqueous  vapor  had  even  nearly  the  same  density  under 
the  same  conditions,  the  presence  of  the  latter  would 
not  affect  the  problem  ;  but  as  watery  vapor  is  only  five 
eighths  as  dense  as  dry  air,  the  weight  of  a  unit  of  vol- 
ume of  the   atmosphere  will  depend  upon  the  amount 
of  vapor  which    it  contains.     Accordingly  a  humidity 
term   has  been   introduced    into   some   barometric   for- 
mulas. 

The  introduction   of  a  humidity  term  requires  that 


ART.  4]  STATICAL    FORMULAS.  337 

the  hygrometric  state  of  the  air  column  shall  be  known, 
and  therefore  an  observation  with  a  hygrometer  must  be 
made  at  each  station.  For  this  purpose  the  wet  bulb 
hygrometer,  or  psychrometer,  is  generally  preferred, 
because  of  its  greater  accuracy  and  convenience.  It 
consists  of  a  pair  of  accurate  thermometers,  one  of 
which  is  exposed  to  the  air  in  the  usual  manner,  while 
the  other  is  exposed  with  a  moistened  bulb.  The 
evaporation  of  moisture  from  the  surface  of  the  wetted 
bulb  has  a  cooling  effect,  and  causes  that  thermometer 
to  indicate  a  lower  temperature  than  the  other. 

Knowing  the  readings  of  the  wet  and  dry  bulb 
thermometers,  the  barometric  pressure  due  to  the 
aqueous  vapor  in  the  air  may  be  determined  from 
tabtes,*  which  are  the  results  of  experiment. 

"In  a  very  general  sense,  in  temperate  climates  near 
the  sea  level,  the  amount  of  vapor  in  the  atmosphere  is 
from  0.2  to  0.4  of  an  inch,  or  about  one  hundredth  of 
the  total  barometric  pressure." 

394.  The  correction  for  humidity  may  be  applied  in 
either  of  two  ways  ;  viz.,  (i)  the  observed  heights  of 
the  barometer  may  be  corrected  for  the  pressure  of  the 
aqueous  vapor  before  substituting  them  in  the  formula; 
or  (2)  the  observed  heights  may  be  used  uncorrected, 
and  the  resulting  altitude  be  multiplied  by  a  factor — the 
humidity  term  of  the  barometric  formula — to  correct 
for  the  effect  of  the  aqueous  vapor  in  the  atmosphere. 
For  formulas  containing  a  humidity  term,  see  §§  401— 
402.  Laplace  slightly  increased  the  co-efficient  of  expan- 
sion of  air  (§  396)  to  partially  compensate  for  the  greater 
expansive  power  of  the  aqueous  vapor.  This  method  of 
correcting  for  humidity  is  incorrect  in  principle,  and 


*  Williamson's  On  the  Use  of  the  Barometer,  Table  C  of  the  Appendix ; 
Smithsonian  Miscellaneous  Collections,  Vol.  I,  Guyot's  Tables,  Group  B,  pp. 
46-72,  and  pp.  102-6;  Lee's  Tables  (3d  ed.),  pp.  172-76. 


338  BAROMETERS.  [CHAP.   XII 

may  at  times  give  rise  to  considerable  error,  particu- 
larly in  case  of  an  extremely  dry  or  an  extremely  moist 
atmosphere,  or  when  the  temperature  of  the  air  is  at  or 
near  32°  F. 

395.  "  It  Is  doubtful  whether  any  considerable  in- 
crease of  accuracy  is  obtained  by  including  a  separate 
correction  for  the  aqueous  vapor.  The  laws  of  the 
distribution  and  transmission  of  moisture  through 
the  atmosphere  are  too  little  known,  and  its  amount, 
especially  in  mountain  regions,  is  too  variable  and 
depends  too  much  upon  local  winds  and  local  condensa- 
tion, to  allow  a  reasonable  hope  of  obtaining  the  mean 
humidity  of  the  layer  of  air  between  the  two  stations 
by  means  of  hygrometrical  observations  taken  at  each 
of  them.  The  observations  for  humidity  are  made  in 
the  stratum  of  air  next  to  the  surface  of  the  earth, 
which  probably  contains  the  greatest  amount  of  moist- 
ure, and  which  is  therefore  least  representative  of  the 
layer  of  air  between  the  two  stations.  At  any  rate,  the 
gain,  if  there  is  any,  is  not  sufficient  to  compensate  for 
the  extra  trouble  in  making  the  observations  and  the 
undesirable  complication  of  the  formula."  * 

At  best  the  determination  of  heights  by  a  barometer 
is  only  an  approximate  method  (§§  368-69),  and  no 
additional  instruments,  nor  any  refinements  of  compu- 
tation, can  make  it  a  comparatively  accurate  method 
of  leveling ;  and  therefore  it  is  not  thought  wise  to 
consider  this  phase  of  the  subject  further. 

396.  TYPICAL  FORMULAS.  Laplace's  Formula.  Laplace 
was  the  first  to  give  a  rational  formula  for  determining 
heights  by  the  barometer,  and  his  formula  has  served 
as  a  basis  for  several  others.  It  is,  for  Fahrenheit  de- 
grees, 


*  Professor  A.  Guyot,  in  Smithsonian   Miscellaneous  Collections,  Vol.  I, 
Art.  Ill,  Group  D,  p.  33. 


ART.  4]  STATICAL    FORMULAS.  339 


X  ft.  =  60158.6  log       < 


(l+:        ^-~— 

(l  -j-  0.002,60  COS  20).  (13) 

,  (l  +  ~2^886,860    +  10,443,435]' 


X  in  the  last  term  is  the  value  of  the  preceding  part  of 
the  formula. 

The    last  term   of  equation  (13)   is  derived  from  the 
corresponding  term  of  equation  (n),  page  332,  as  fol- 

lows  :     Place  lo       i  +  -  =  0.4    .  .  ---.     Find 


log   (i  +  -^  J  =  0.43  .  .  - 


TT 

log  —  \  by  omitting  the  terms  in  equation  (u)  after  the 

brace  and  solving,   calling  5.744  —  -  =  60,158.       Substi- 

tute these  results  in  the  last  term  of  equation  (n),  per- 
form the  operations  indicated,  and  omit  the  terms  con- 
taining R*  in  the  denominator.  See  §  392. 

The  true  co-efficient  of  expansion  of  air  is  0.003,75 
per  i°  C,  but  Laplace  increased  it  to  0.004  "  to  allow 
for  the  greater  expansive  force  of  the  aqueous  vapor  in 
the  atmosphere."  *  See  §§  394  and  395. 

The  second  term  in  the  last  line  of  equation  (13)  is 
usually  called  "  the  correction  for  the  variation  of  grav- 
ity on  the  mercury,"  and  the  third  term  in  the  same 
line  "  the  correction  for  the  variation  of  gravity  on  the 
air";  but  an  examination  of  the  method  by  which  the 
last  term  was  deduced  will  show  that  this  explanation 
of  these  terms  is  erroneous. 

Numerous  tables  have  been  prepared  to  facilitate  the 
application  of  the  above  formula  ;  for  example,  see 


*  Prof.  A.  Guyot  in  Smithsonian  Miscellaneous  Collections,  Vol.  I,  Art. 
Ill,  Group  D,  p.  9. 


340  BAROMETERS.  [CHAP.   XII 

Smithsonian  Miscellaneous  Collections,  Vol.  I,  Art.  Ill, 
Group  D,  Guyot's  Table,  pp.  35-48  ;  ibid.t  Loomis's 
Table,  pp.  49-53  ;  Loomis's  Practical  Astronomy,  pp. 
390-91. 

Tables  are  useful  where  a  great  number  of  observa- 
tions are  to  be  reduced;  but  they  generally  contain  an 
unnecessary  number  of  figures,  and  hold  forth  a  show 
of  extreme  accuracy  which  the  nature  of  the  observa- 
tions can  not  justify. 

397.  Babinet's  Formula.  Babinet's  formula*  for  Fahren- 
heit degrees  is 


X  ft.  =  60,334  log         1  +  .      (I4) 


Notice  that  this  formula  has  no  term  for  the  variation 
of  gravity.  It  is  sometimes  claimed  f  that  the  baromet- 
ric co-efficient  is  Laplace's  (§  396)  increased  to  include 
the  correction  due  to  the  variation  of  gravity  with  the 
altitude;  but  owing  to  the  nature  of  the  case  this 
procedure  is  entirely  indefensible 

Searle's  Field  Engineering  (p.  4  and  pp.  307-9)  con- 
tains tables  to  facilitate  the  application  of  this  formula. 

398.  Babinefs  Approximate  Formula.  If  H'  and  Hv  do 
not  greatly  differ  it  can  readily  be  shown  that 

H          H  —  H  ,.. 


in  which  M  is  the  modulus  of  the  common  system  of 
logarithms.  Making  this  substitution  in  equation  (14) 
gives  Babinet's  approximate  formula, 


*  Guyot's  Tables,  Group  D,  p.  68. 

t  Prof.  A.  Guyot  in  Smithsonian  Miscellaneous  Collection,  Vol.  I,  Art,  III, 
Group  D,  p.  9. 


ART.  4]  STATICAL    FORMULAS.  341 

The  mathematical  approximation  involved  in  equa- 
tion (15)  is  inappreciable  for  elevations  less  than  3,000 
feet. 

399.  The  following  formula*  is  essentially  the  same 
as  Babinet's  approximate  formula — equation   (16) — ex- 
cept  the  value   of  the  barometric  co-efficient  and  the 
omission    of    the  temperature   term.     Notice    that   the 
barometric  constant  of  equation  (17)  is  larger  than  that 
of  equation  (16),  which  was  derived  from  a  relatively 
large    value,   i.e.,    that    of    equation    (14).     A    similar 
formula     is    freqr^ntly    given    with    slightly    different 
constants ;    for   example,   see    Lee's    Tables   (3d   ed.), 
p.  151. 

TT      _          TTt  •}£ 

X  ft.  =  54,500^  +  ff,  ±  —  ±  10.    .    .    .    .     (17) 

The  last  two  terms  of  equation  (17)  are  supposed  to 
show  the  degree  of  reliance  to  be  placed  upon  the 
result.  Formulas  (16)  and  (17)  are  much  used  with 
"compensated"  aneroid  barometers  (see  §§  342-43). 

400.  Bailey's  Formula.     All    the   preceding   formulas 
require  that  the  barometric  reading  be  reduced  to  32°  F. 
before  being  inserted  in  the  formula;  but  in   Bailey's 
formula,  equation  (i8),f  the  readings  are  used  without 
any  reduction  (see  §  389). 


X  ft.  =60,346  log  h-\ 


h*      i  +  o.oooi^j  —  /) 

r  +  r,  -.jS4 

(l  +  O.OO2,695    ^^    20) 


*  U.  S.  Coast  Survey  Report,  1876,  pp.  352-53. 
t  Guyot's  Tables,  Group  D,  p.  69. 


342  BAROMETERS.  [CHAP.  XII 

in  which  hl  and  h*  are  the  observed  readings  of  the 
barometers  at  the  two  stations,  and  tl  and  /'  the  tem- 
peratures of  the  barometer  in  Fahrenheit  degrees. 

Tables  for  the  application  of  Bailey's  formula  are 
given  in  Lee's  Tables  (2d  ed.),  pp.  84,  85,  and  in 
Smithsonian  Miscellaneous  Collections,  Vol.  I,  Art.  Ill, 
Group  D,  pp.  70,  71. 

401.  Plantamour's  Formula.  None  of  the  preceding 
formulas  takes  account  of  the  humidity  of  the  at- 
mosphere, except  by  the  erroneous  method  of  increasing 
the  co-efficient  of  expansion  of  air  (§  396).  Bessel  was 
the  first  to  propose  a  separate  humidity  term.  Planta- 
mour's formula,*  which  differs  from  the  one  proposed 
by  Bessel  only  in  the  form  and  the  value  of  the  con- 
stants, is 

X  (in  meters)  =  log  H^  -  log  H'  +  V 

+ — V  -r+*;  to) 

I-JF4±4r 


in  which 


=          398.25  jg, 

397.25  -  cT    a0 


30.030,197,57-  -  0.000,080,1701™ 


397.25  -  cT 
q  =  i  —  0.002,625,7  cos  0, 

s  and  $'  =  the  fraction  of  saturation  of  the  layer  of  air 
between  the  two  statrons. 

Tables  for  the  application  of  this  formula  are  given  in 
Smithsonian  Miscellaneous  Collections,  Vol.  I,  Guyot's 
Tables,  Group  D,  pp.  78,  79. 

*  Guyot's  Tables,  Group  D,  p.  75. 


AkT.  4]  STATICAL   FORMULA^  343 

402.  Williamson's  Formula.  Williamson's  formula,* 
equation  (20),  is  a  translation  of  Plantamour's  formula 
into  Laplace's  form. 

X  ft.  =  60,384  log  -j±f 

7^4.7* -64 

982.2647 

(l   —  0.002,625,7  COS  20) 

z      x        (20) 


20,886,860  10,443,4307 


((i+M) 


in  which  (i  -j-  ^O  'ls  tne  humidity  term  (§§  393-95). 

Williamson's  On  the  Use  of  the  Barometer  —  pages 
111-55  °f  the  Appendix  —  contains  elaborate  tables  for 
the  application  of  this  formula.  The  same  tables  are 
also  given  in  Lee's  Tables  (3d  ed.),  pp.  152-71. 

403.  Altitude  Scales  of  Aneroids.  The  dials  of  many 
aneroid  barometers  have  a  scale  engraved  upon  them,  by 
which  the  elevation  is  read  directly.  The  scale  may  be 
graduated  according  to  any  of  the  preceding  formulas, 
but  apparently  equation  (17),  page  341,  is  most  frequently 
employed  for  this  purpose.  Sometimes  the  zero  of  the 
altitude  scale  is  placed  to  correspond  with  a  pressure  of 
30  inches  of  the  mercurial  barometer,  as  though  the 
scale  of  elevations  could  be  employed  to  determine  ab- 
solute elevations.  But,  obviously,  the  altitude  scale  can 
not  give  absolute  elevations  with  any  considerable  ac- 
curacy, owing  to  the  variations  in  pressure,  temperature, 
and  humidity  (§§349-61);  and  consequently  it  should 
be  used  only  to  find  differences  of  elevations,  in  which 
case  the  difference  of  level  is  obtained  by  subtracting 

*  Williamson's  On  the  Use  of  the  Barometer,  p.  102  of  the  Appendix, 


344  BAROMETERS.  fcHAP.  XII 

the  reading  in  feet  at  the  lower  station  from  that  at  the 
upper.  To  prevent  a  misuse  of  the  aneroid  in  this  re- 
spect, the  zero  of  the  altitude  scale  is  sometimes  placed 
to  correspond  with  31  inches  of  pressure,  which  gives 
such  large  numbers  as  to  suggest,  particularly  when  find- 
ing small  elevations,  the  proper  use  of  the  instrument. 

Sometimes  the  altitude  scale  is  engraved  upon  a  mov- 
able ring  encircling  the  dial,  so  that  the  scale  may  be 
set  to  agree  with  the  known  altitude  of  any  station  (see 
§  343,  paragraph  7)  ;  and  then,  as  the  instrument  is 
carried  about,  the  pointer  will  indicate  with  a  fair  de- 
gree of  approximation,  for  some  hours,  the  altitude  of 
stations  at  which  it  is  read.  This  device  is  convenient, 
and  in  the  hands  of  an  intelligent  observer  who  requires 
rapid  work  and  desires  only  approximate  results,  it  is  a 
valuable  modification;  but  such  a  use  of  the  movable 
scale  may  at  times  involve  large  errors,  as  it  is  based 
on  the  assumption  that  differences  of  pressure  cor- 
respond at  all  heights  with  the  same  differences  of  ele- 
vation. 

404.  The  altitude  scale  can  be  correct  only  for  some 
particular  temperature  of  the  air;  and  consequently,  if 
the  most  accurate  results  are  desired,  a  correction  for 
the  temperature  of  the  air  (§  390)  must  be  applied. 
With  some  instruments  this  correction  is  made  by  mov- 
ing the  altitude  scale.  In  this  case  the  scale  is  graduated 
for  some  particular  temperature,  its  zero  is  adjusted  by 
reading  the  instrument  at  some  station  of  known  eleva- 
tion, and  this  position  of  the  zero  is  marked  by  aline  on 
the  dial  numbered  to  correspond  with  the  normal  tem- 
perature; then  points  are  determined — to  quote  the 
words  of  the  inventor  of  this  method  of  applying  this 
correction, — "  partly  by  trial  and  partly  by  computa- 
tions," at  which  the  zero  of  the  altitude  scale  should  be 
set  for  different  temperatures  of  the  air.  With  this  form 
of  instrument,  the  scale  must  be  set  for  the  mean 


ART.  4]  DYNAMICAL    FORMULAS.  345 

temperature  of  the  air,  before  any  readings  are  taken; 
and  it  must  not  be  shifted  during  the  progress  of  the 
work. 

The  method  of  applying  the  correction  for  the  tem- 
perature of  the  air  by  shifting  the  altitude  scale,  is  not 
theoretically  correct;  and  the  accuracy  of  the  results 
depend  entirely  upon  the  accuracy  of  the  scale  by  which 
the  altitude  scale  is  shifted.  The  best  plan  is  to  dis- 
pense entirely  with  altitude  scales,  whether  fixed  or 
movable,  and  calculate  the  heights. 

405.  Conclusion.     The  preceding  do  not  comprise  all 
the  formulas  which  have  been  proposed  for  barometric 
leveling,  but  include  the  more  common  ones,  and  illus- 
trate all  the  principles  involved — except  those  discussed 
in    §§406-9.      Some   of   the    omitted  formulas   are  ap- 
proximate, some  have  empirically  determined   pressure 
co-efficients,  etc.     Owing   to    the  matters  discussed   in 
§§349-67  it  is   comparatively  unimportant  which  of  the 
generally  recognized  barometric  formulas  is  used. 

B.  Dynamical  Formulas. 

406.  FerreFs  Formula.     Ferrel  has  deduced  a  formula* 
which  is  claimed  to  be  independent  of  the  defects  of  all 
formulas  founded  upon  an  assumed  statical  condition 
of  the  atmosphere.     Although  the  formula  is  very  care- 
fully and  ingeniously  worked  out,  it  is  probably  of  little 
use  for  ordinary  hypsometrical  work,  since  it  requires 
observations  to  be  made  for  a  long  time  over  a  consid- 
erable area,  to  get  the   data  by  which  to  compute  cor- 
rections   for    gradient,    temperature,     and     humidity. 
Without  the  data   for    making   these    reductions,    this 
formula  is  essentially  the  same  and  has  essentially  the 


*  Report  of  U.  S.  Coast  and  Geodetic  Survey,  1881,  pp.  225-68,  the  for- 
mula  itself  being  given  on  page  235. 


346  BAROMETERS.  [CHAP.  XII 

same   defects    as    the    formulas    depending   upon    an 
assumed  statical  condition  of  the  atmosphere. 

407.  Gilbert's  Formula.  This  formula,  unlike  all  those 
in  division  A  (§§  383-405),  does  not  require  the  observa- 
tion of  either  the  temperature  or  humidity  of  the  air; 
but  requires  observations  of  the  barometer  at  three 
stations,  the  difference  of  elevation  of  two  of  which  is 
known  and  the  elevation  of  the  third  of  which  is  to  be 
determined.  The  formula  is 


,  .  . 

log/—  log  u          490,000 

in  which  B  is  the  difference  in  elevation  of  the  two 
stations,  /  is  the  reading  of  the  barometer  at  the  lower 
station,  reduced  to  32°  F.,  u  the  same  for  the  upper  sta- 
tion, and  n  the  same  for  the  station  whose  elevation  is 
to  be  determined,  and  A  is  the  value  of  the  first  term 
of  the  second  member.  This  formula  is  deduced  on 
the  assumption  that  the  three  stations  are  in  the  same 
vertical,  and  also  that  the  station  whose  altitude  is  to  be 
determined  is  between  the  other  two.  The  less  nearly 
these  conditions  are  fulfilled,  the  more  nearly  this 
formula  is  liable  to  the  same  errors  as  statical  formu- 
las. The  denominator  of  the  last  term  was  determined 
by  experiment,  and  is  therefore  liable  to  error.  See 
§33o. 

408.  Weilenmann's  Formula.  The  Report  of  the  U. 
S.  Coast  and  Geodetic  Survey  for  1876,  pages  388-90, 
contains  a  comparatively  brief  discussion  of  a  baromet- 
ric leveling  formula  based  upon  the  dynamical  theory 
of  heat.  In  the  one  case  to  which  this  formula  is  ap- 
plied, it  gives  errors  one  half  less  than  the  best  of  the 
ordinary  barometric  formulas;  but  as  the  errors  in  the 

*  G.  K.  Gilbert  in  U.  S.  Geological  Survey  Report  for  1880-81,  p.  448. 


ART.  4]  DYNAMICAL   FORMULAS.  347 

single  example  to  which  it  is  applied,  due  to  diurnal 
variations  of  pressure,  amount  to  one  per  cent,  and  as  it 
is  much  more  complicated  than  the  ordinary  formulas, 
it  will  not  be  considered  further. 

409.  Conclusion.  Owing  to  the  possibility  of  errors 
due  to  gradient,  temperature,  and  humidity  (§§  348-67), 
there  is  but  little  difference  in  accuracy  between  statical 
and  dynamical  formulas.  See  §  381. 


APPENDIX  I. 

ELIMINATION  OF   LOCAL  ATTRACTION  IN 

SURVEYING  WITH  THE  MAGNETIC 

COMPASS. 

1.  IT  is  very  common   to  hear  the  remark  that  the 
magnetic  compass  can  not  be  used  because  of  local  at- 
traction.    It  is  welt  known  that  there  are   many  locali- 
ties in  which   the  needle  is  deflected  by  the  attraction 
of  iron,  etc.;  but  the  object  of  this   article  is  to  show 
that  no  matter  how  much  the  local  deflection,  the  com- 
mon magnetic  compass  can  be  made  to  give  as  accurate 
results  as  though  there  was  no  disturbing  influence. 

Since  the  conditions  for  mine  surveying  differ  slightly 
from  those  for  land  surveying,  this  subject  will  be  consid- 
ered under  the  heads  of  Mine  Surveys  and  Land  Surveys. 

ART.  1.     MINE  SURVEYS. 

2.  Generally  a  survey  of  a  mine  is  made  to  determine 
the   relative    position    of    the    underground    workings 
and    the   boundary  lines  of  surface   property.     Conse- 
quently it  is  necessary  to  find  in  the  mine  (i)   a  point 
directly  under  a  known  point  on  the  surface,  and  (2) 
the  direction  of  a  line  corresponding  to  a  known  line 
on  the  surface.     The  first  may  be  found  by  means  of  a 
plumb-line,  or  its  equivalent;  and  the  second  by  the  use 
of  two  plumb-lines,  or  by  a  transit — usually  the  former. 
For  the  present  it  will  be  assumed  that  the  true  direc- 
tion of  the  initial  underground  line  is  known. 

For  the  sake  of  illustration,  assume  that  a  survey  is 

349 


35° 


ELIMINATION    OF    LOCAL    ATTRACTION. 


to  be  made,  starting  from  some  point,  say  A,  on  a  line 
whose  direction  is,  say,  S.  78°  10'  E.  Set  the  compass 
at  Ay  and  also  at  each  corner  or  station  of  the  survey, 
successively,  and  read  the  bearing  of  the  two  lines 
meeting  in  each  station.  Call  the  sight  made  in  the 
direction  in  which  the  survey  progresses  a  fore-sight, 
and  that  made  in  an  opposite  direction  a  back-sight. 
Keep  one  end  of  the  box  next  to  the  eye  on  fore-sights 
and  the  other  end  on  back-sights.  But  if  one  sight  of  the 
compass  consists  of  a  slit  and  the  other  of  a  hair,  the 
same  end  must  necessarily  be  kept  next  to  the  eye; 
therefore  read  the  letters  from  one  end  of  the  needle 
for  fore-sights,  and  from  the  opposite  end  for  back- 
sights, but  the  degrees  from  the  same  end  all  the  time. 
When  all  the  bearings  have  been  taken,  the  record  will 
be  similar  to  the  first  three  columns  of  the  following 
table,  with  the  exception  of  the  small  figures  written 
above  each  bearing,  which  will  be  explained  presently. 

TABLE   I. 


Station. 

Back-sight. 

Fore-sight. 

Correction. 

78     10 

16     05 

A 

S.  78°  oo'  E. 

N.  15°  55'  W. 

io'B. 

16      05 

80    30 

Q 

N.  16°  45'  W. 

N.  81°  10'  W. 

40'  F. 

80    30 

46      10 

s 

N.  80°  05'  W. 

S.  46°  35'  W. 

25'  B. 

46      10 

83     oo 

u 

S.  46°  35'  W. 

S.  82°  35'  E. 

25'  B. 

83      oo 

78      oo 

V 

S.  82°  25'  E. 

S.  77°  25'  E. 

35'  B. 

As  explained  above,  the  true  bearing  of  the  initial  or 
reference  line  is  S.  78°  10'  E.,  and  hence,  if  there  had 
been  no  local  attraction  at  A,  the  back-sight  at  that  sta- 
tion, i.e.,  the  bearing  of  the  reference  line,  would  have 
been  S.  78°  10'  E.  But  as  it  read  S.  78°  o'  E.,  there 
is  a  local  attraction  of  10',  and  therefore  the  bearings 


ART.    l]  MINE    SURVEYS.  351 

taken  at  A  are  in  error  10'.  The  needle  read  S.  78°  o' 
E.,  but  if  there  had  been  no  local  attraction  it  would 
have  read  S.  78°  10'  E.  The  true  reading  is  therefore 
10'  farther  from  the  south  toward  the  east  than  the 
observed  reading,  and  hence,  to  get  the  true  bearing 
the  needle  should  be  moved  10'  in  a  direction  from  the 
south  towards  the  east.  The  fore-sight  at  A  is  N.  15° 
55'  W.;  but  to  get  correct  bearings  at  this  station,  we 
must,  in  imagination,  move  the  needle  10'  in  a  direction 
from  the  south  towards  the  east,  which  is  the  same 
direction  as  from  the  north  towards  the  west,  and  hence, 
the  correct  bearing  of  the  line  starting  from  A  is  N.  15° 
55'  W.,  plus  10',  which  gives  N.  16°  5'  W. 

For  simplicity  of  explanation,  let  us  call  a  direction 
from  the  south  towards  the  west  forward,  and  the  op- 
posite direction  backward,  these  terms  being  assigned 
to  agree  with  the  direction  of  the  motion  of  the  hands 
of  a  watch.  It  makes  no  difference  in  this  matter 
whether  we  consider  N.,  S.,  E.,  and  W.  in  their  true 
relations,  or  in  their  reversed  position  as  given  by  the 
face  of  the  compass.  We  will  assume  them  to  be  in 
their  true  relations,  and  briefly  say  that  the  correction 
at  A  is  10'  backward.  This  is  the  correction  which  is 
to  be  applied  to  the  fore-sight  at  A,  and  is  written  in 
the  column  headed  "  correction."  The  true  bearing 
of  the  initial  line  is  written  over  the  back-sight  taken 
at  A,  and  the  corrected  fore-sight  is  written  above  the 
fore-sight  observed  at  Q. 

At  Q  the  back-sight,  />.,  the  bearing  of  the  line  from 
A  towards  Q,  is  N.  16°  45'  W.;  but  as  the  true  bearing  is 
N.  16°  5'  W.,  as  found  for  the  corrected  fore-sight  at  A, 
the  error  at  Q  due  to  local  attraction  is  40'.  The  cor- 
rection at  Q  is  therefore  40'  forward,  and  the  corrected 
fore-sight  is  80°  30'.  The  corrections  and  corrected 
bearings  for  the  other  stations  are  found  in  a  similar 
manner, 


352 


ELIMINATION    OF    LOCAL    ATTRACTION.      [APPEN.   I 


3.  As  is  easily  seen,  when  the  true  bearing  of  the 
initial  line  is  known,  the  above  method  absolutely 
eliminates  all  local  attraction.  On  the  other  hand,  if 
the  direction  of  the  initial  line  is  not  known,  this 
method  will  give  a  fairly  good  determination  of  it.  To 
find  the  true  bearing  of  all  of  the  lines,  including  that 
of  the  first  one,  set  the  compass  at  each  station,  and 
read  the  back-sights  and  fore-sights  at  each  as  in  the 
preceding  case.  The  record  will  then  be  as  in  Table  II, 
except  for  the  small  figures  above  the  bearings. 

TABLE   II. 


Station. 

Back-sight. 

Fore  sight. 

Correction. 

A 

78         10 

S.  77°  50'  E. 

16     05 

N.  i5°45'W. 

20'  B. 

Q 

16     05 

N.  1  6°  35'  W. 

80     30 
N.  81°  oo'  W. 

30'  F. 

S 

N.  80°  30'  W. 

S.  46°  10'  W. 

0 

U 

S.  46°  10'  W. 

S.  83°  oo'  E. 

o 

V 

S.  83°  oo'  E. 

S.  78°  oo'  E. 

0 

w 

78      oo 

S.  78°  10'  E. 

N.  85°  25'  E. 

10'  F. 

Notice  that  the  fore-sight  at  S  agrees  with  the  back- 
sight at  Uj  and  also  that  the  fore-sight  at  £7  agrees  with 
the  back-sight  at  V.  This  makes  it  extremely  probable 
that  there  is  no  local  attraction  at  these  three  stations, 
and  that  the  correction  at  these  stations  is  o°.  To  find 
the  correction  at  Q,  write  the  back-sight  at  S  above  the 
fore-sight  at  Q,  and  take  the  difference  between  the  ob- 
served and  the  corrected  fore-sight.  This  difference  is 
30',  and  is  the  correction  at  Q.  A  moment's  reflection 
shows  that  this  correction  is  "  forward,"  as  marked. 
The  remaining  bearing  at  Q  and  the  two  at  bearings  A 


ART.   l] 


MINE    SURVEYS. 


353 


may  be  found  as  already  explained,  and  similarly  those 
at  W\  that  is  to  say,  the  bearings  in  the  upper  part  of 
the  table  are  corrected  by  working  up  from  S,  and  those 
in  the  lower  part  by  working  down  from  V. 

4.  It  may  happen  that  not  even  two  successive  stations 
are  found  at  which  the  corresponding  back-sight  and 
fore-sight  agree.  When  this  occurs,  assume,  for  tempo- 
rary purposes,  that  the  correction  at  the  first  station  is 


zero,  and   correct  all   the    bearings  accordingly, 
result  will  then  be  as  in  Table  III. 

TABLE  III. 


The 


Station. 

Back-sight. 

Fore-sight. 

Correction. 

A 

S.  78°  00'  E. 

N.  75°     5'  E. 

0 

75      55 

80        o 

P 

N.  75'  35'  E. 

N0  80°  LO'  E. 

20'  F. 

80        20 

85    30 

R 

N.  80°  10'  E. 

S.  85°  40'  E. 

lo'F. 

85    30 

88      50 

T 

S.  85°  50'  E. 

N.  88°  30'  E. 

20'  F. 

88      50 

78      oo 

X 

N.  88°  20'  E. 

S.  78°  30'  E. 

30'  F. 

78      oo 

85         20 

Y 

S.  78°  20'  E. 

N.  85°  oo'  E. 

20'  F. 

Notice  that  there  are  three  stations  at  which  the  cor- 
rection is  the  same,  i.e.,  20'  F.  This  indicates  that  the 
apparent  local  attraction  at  these  stations  is  the  same;  it 
is  therefore  probable  that  the  needle  had  its  normal  or 
natural  position  at  these  stations,  and  that  the  local  at- 
traction was  confined  wholly  to  the  other  stations.  If 
this  conjecture  be  correct  (its  validity  will  be  examined 
farther  presently),  then  the  corrections  at  P,  T,  and  Y 
are  zero.  By  inserting  this  correction  at  these  stations, 
and  correcting  the  notes  as  previously  explained,  we  get 
Table  IV,  which  gives  the  most  probable  bearings  of  the 
several 


354 


ELIMINATION    OF    LOCAL    ATTRACTION.    [APPEN.  I 


5.  For  the  preceding  illustration  the  conjecture  is 
slightly  defective,  since  the  agreement  of  the  corrections 
at  only  three  stations  might  possibly  be  due  to  an  acci- 


TABLE  IV. 


Station. 

Back-sight. 

Fore-sight. 

Correction. 

A 

78     20 
S.  78°  oo'  E. 

N.  775°  535'  E. 

20'  B. 

P 

R 

N.  75°  35'  E. 
N.  80°  i°o'  E. 

N.  80°  oo'  E. 

85     50 
S.  85°  40'  E. 

o 
10'  B. 

T 
X 

S.  85°  50'  E. 

88      30 

N.  88°  20'  E. 

N.  88°  30'  E. 

78          20 

S.  78°  30'  E. 

o 
lo'F. 

Y 

S.  78°  20'  E. 

N.  85°  oo'  E. 

o 

dental  agreement  of  the  local  attraction  at  these  stations. 
If  this  is  the  case,  then  it  is  impossible  by  this  method 
to  find  the  true  bearings  of  the  lines,  unless  proportion- 
ally more  stations  can  be  found  at  which  the  apparent 
local  attractions  agree.  For  example,  if  the  next  five 
stations  give  an  apparent  local  attraction  agreeing  with 
that  found  in  Table  III  for  station  X,  then  those  stations 
must  be  regarded  as  having  the  normal  position  of  the 
needle,  and  the  bearings  of  the  others  must  be  corrected 
accordingly.  On  the  other  hand,  if  it  is  impossible  to 
find  a  number  of  stations  at  which  the  apparent  local 
attractions  agree,  it  is  impossible  to  determine  the  true 
bearings  with  the  needle  alone.  What  then  ? 

If  the  true  direction  of  the  initial  line  has  not  been 
determined  by  plumbing  the  shaft,  and  if  the  apparent 
local  attraction  does  not  agree  at  a  proportionally  large 
number  of  stations,  observe  the  back-sights  and  fore- 
sights as  previously  described,  and  correct  the  bearings 
with  reference  to  the  initial  line  as  explained  in  Table 
III.  Of  course  this  method  does  not  certamiy  deter- 


ART.    l]  MINE    SURVEYS.  355 

mine  the  true  bearings,  but  it  finds  correctly  the  relative 
directions  of  the  several  lines,  and  all  the  computations 
and  plats  will  be  perfectly  correct,  except  that  they  will 
be  turned  around  an  amount  equal  to  the  difference 
between  the  assumed  and  the  true  bearing  of  the  initial 
line.  If  the  survey  is  made  to  determine  the  position  of 
the  various  parts  of  the  mine  relative  to  the  land  lines 
above,  then  in  many  cases,  particularly  in  the  coal  mines 
of  Illinois,  this  method  would  give  fairly  good  results, 
since  the  error  would  probably  be  slight,  while  the 
value  of  the  vein  would  be  nearly  uniform.  Farther,  if 
at  any  subsequent  time  the  true  direction  of  the  initial 
line  should  be  determined,  then  the  correct  bearings  of 
all  the  lines  of  the  underground  survey  could  be  deter- 
mined at  once  by  applying,  as  a  "correction,"  the  dif- 
ference between  the  true  and  the  assumed  direction  of 
the  first  line. 

6.  Finally,  it  may  happen  that  the  apparent  local  at- 
tractions as  found  by  the  trial  balance,  Table  III,  pre- 
ponderate in  one  direction.  For  example,  in  Table  III 
it  will  be  noticed  that  all  the  corrections  are  "  forward  " 
except  that  at  A.  As  a  general  rule  local  attraction  is 
as  likely  to  be  in  one  direction  as  another,  and  hence, 
unless  the  local  conditions  furnish  some  good  reason  to  the 
contrary ',  the  corrections  found  by  the  trial  balance 
should  be  about  as  much  in  one  direction  as  the  other, 
/>.,  the  sum  of  the  "forward"  corrections  should  be 
nearly  equal  to  the  sum  of  the  "  backward  "  corrections. 
This  is  not  so  in  Table  III,  which  is  an  actual  survey. 
The  local  conditions  gave  no  indications  that  the  local 
attraction  was  more  in  one  direction  than  another,  and 
we  may  reasonably  assume  that  the  corrections  are  all 
one  way  in  the  table  because  the  true  correction  &\.A  is 
backward  an  amount  about  equal  to  the  average  of  the 
corrections  at  the  other  stations,  i.e.,  about  18'.  If  this 
correction  were  applied  to  the  original  bearings  in 


356  ELIMINATION    OF    LOCAL    ATTRACTION.     [APPEN.  I 

Table  III,  the  resulting  directions  would  be  nearly  the 
true  ones;  but  if  the  correction  at  any  station  is  con- 
siderably larger  than  any  of  the  others,  it  is  evident 
that  the  local  attraction  at  this  station  is  considerably 
more  than  the  average,  and  hence  this  method  of  cor- 
recting the  bearings  should  not  be  employed, — at  least 
the  excessive  correction  should  not  be  included  in  find- 
ing the  average. 

In  Art.  2  a  modification  of  the  preceding  methods  will 
be  considered,  which,  under  certain  conditions,  may  be 
of  great  importance  in  making  mine  surveys. 

ART.  2.     LAND  SURVEYS. 

7.  Land  surveys  differ  from  mine  surveys  in  that  the 
former   close,  i.e.,    return    to    the    point    of    beginning, 
while  the  latter,  as  a  rule,  do   not.     When  the  survey 
closes,  the  above  method  of  observing  and  correcting 
the  bearings  possesses,  in  addition  to  finding  the  true 
bearings    as   explained    above,    two    important    advan- 
tages :    i,  it  affords  under  every  condition  the  means 
of  checking  the  accuracy  of  the  bearings  ;  2,  it  enables 
the  true  area  to  be  determined,  even  if  the  true  bearing 
of  the  sides  can   not  be  found.     There  are  then   two 
cases,  I,  when  only  the  area  is  wanted  ;  II,  when  the 
area  and  also  the  true  bearings  are  required. 

CASE  I.      When  only  the  area  is  wanted. 

8.  The  angles  are  read   as  described  in  Art.   1,  the 
results  being  as  in  Table  V  (page  357). 

Since  the  assumption  is  that  only  the  area  is  required, 
it  is  immaterial,  where  we  commence  to  correct  the 
angles  ;  and  therefore  we  may  assume  that  the  bearings 
taken  at  any  station,  say  Z,  are  correct,  and  that  the 
correction  at  Z  is  zero.  Beginning  then  at  Z,  we 
may  correct  the  bearings  by  the  method  already  ex- 


ART.   2] 


LAND    SURVEYS. 


357 


plained.  It  is  immaterial  whether  the  angles  are  cor- 
rected in  the  order  in  which  they  were  surveyed,  or  the 
opposite. 

TABLE  V. 


Stations. 

Back-sight. 

Fore-sight. 

Correction. 

z 

S.  21°  00'  W. 

N.    6°  55'  W. 

0 

6       55 

55      25 

Y 

N.    7°os'W. 

N.  55°  35'  W. 

10'F. 

55      25 

36      3° 

X 

N.  55°  35'  W. 

S.  36°  20'  W. 

10'  F. 

36    30 

2        2O 

W 

S.  36°  50'  W. 

S.    2°  oo'  E. 

20'  B. 

2         20 

43      °° 

u 

S.    2°  40'  E. 

S.  43°  20'  E. 

20'  F. 

43      °° 

69     25 

T 

S.  43°  05'  E. 

S.  69°  30'  E. 

5'F. 

69    25 

21        5 

S 

S.  69°  30'  E. 

N.  21°  oo'  E. 

5'F. 

It  will  be  noticed  in  the  example  cited  that  the  back- 
sight at  Z  differs  from  the  corrected  fore-sight  at  S.  If 
the  angles  had  been  correctly  read,  these  two  would 
have  agreed  exactly.  The  difference  between  the  first 
corrected  back-sight  and  the  last  corrected  fore-sight 
shows  the  error  of  reading  the  angles.  This  difference 
should  not  exceed  5'  or  10'  (see  §  50,  page  52).  (The 
example  in  Table  V  is  an  actual  survey  made  by  one  of 
the  author's  students.) 

CASE  II.      When  the  area  and  also  the  true  bearings  are 

required. 

9.  This  case  requires  one  of  two  things  :  either  that 
several  successive  back-sights  shall  agree  with  their 
corresponding  fore-sights  before  either  have  been  cor- 
rected, or  that  there  shall  be  a  true  meridian  which  can 
be  connected  with  a  corner  of  the  field  by  lines  whose 


358 


ELIMINATION    OF    LOCAL    ATTRACTION.    [APPEN.  I. 


bearings   are  to  be  found  in  the  same  manner  as  are 
those  of  the  boundary  lines. 

10.  In  the  first   instance  it  may    safely  be  assumed 
that,  since  a  number  of  the  fore-sights  and  back-sights 
agree,  the  observed  bearings  are  the  true  ones.     Hav- 
ing some  of  the  correct  bearings,  if  there  are  any  sta- 
tions  at  which  the  back-sights   do  not  agree  with  the 
corresponding  fore-sights,  they  may  be  corrected  as  in 
Table  III.     If  the  declination  is  not  set  off  on  the  com- 
pass, it  may  be  applied  as  a  correction  to  those  stations 
at   which    the    back-sight  and  fore-sight  corresponded 
as  above,  and  the  correction  may  then  be  carried  on  to 
the  remaining  stations. 

11.  In  the  second  instance,  the  true  bearings  of  the 
several   lines  can  be  found  in  succession,  beginning  at 
the  meridian.     The  example  in  Table  VI  will  illustrate 


TABLE  VI. 


Stations. 

Back-sight. 

Fore-sight. 

Correction. 

Z 

73      10 

N.  69°  20'  E. 

3°  50'  F. 

Q 

73     I0 

N.  69°  15'  E. 

3°  55'  F. 

Q 

14    40 
N.  10°  45'  E. 

93      4° 

S.  89°  45'  W. 

3°  55'  F. 

T 

93      4° 

S.  89°  35'  W. 

43      °5 

S.  39°  oo'  W. 

4°  05'  F. 

U 

43      05 

S.  39°  10'  W. 

M      35 

N.  10°  40'  E. 

3°  55'  F. 

this  case.  Z  is  not  a  corner  of  the  field,  but  a  point  on 
a  true  meridian,  the  line  ZQ  being  run  to  determine 
the  declination  at  Q,  a  corner  of  the  field.  All  the  re- 
maining stations  are  corners  of  the  field. 

Notice  that  the  first  two  lines  of  the  above  record  are 
preliminary  to  the  survey  of  the  field,  and  are  required 


ART.  2]  LAND    SURVEYS.  359 

only  to  find  the  declination  or  correction  at  Q.  Hav- 
ing found  the  correction  at  Q,  the  bearings  are  cor- 
rected as  in  the  previous  case.  The  back-sight  from 
Q  to  ^/differs  5'  from  the  fore-sight  from  £7  to  Q,  which 
shows  that  there  was  an  error  or  inaccuracy  of  5'  in 
reading  the  angles. 

12.  Undoubtedly  the  method  of  connecting  with  a 
meridian  is  more  exact  than  the  previous  one,  although 
the  former  is  more  convenient  and  shorter,  and  is  the 
one  which  will  generally  be  used  in  practice.  However, 
if  the  bearings  show  local  attraction  at  a  number  of 
stations,  the  first  method  of  Case  II  can  not  be  applied, 
while  the  second  will  give  strictly  correct  results  what- 
ever the  number  of  stations  at  which  local  attraction 
exists. 

Notice  that  when  the  survey  closes,  the  method  ex- 
plained above  absolutely  eliminates  all  local  attraction, 
and  also  that  this  method  is  frequently  applicable  in 
mine  surveying.  Finally,  the  back-sight  should  always 
be  taken,  even  though  it  is  probable  that  no  local 
attraction  exists,  since  the  back-sights  afford  a  very 
valuable  means  of  checking  the  readings  of  the  needle. 


APPENDIX   II. 

FINDING   AREA   BY  TRAVERSING   WITH  THE 
TRANSIT. 

1.  TAKING  THE  FIELD  NOTES.    Assume  that  not  only 

the  area  is  desired,  but  also  the  angles  which  the 
several  sides  make  with  the  meridian.  Place  the  instru- 
ment over  the  first  station,  say  A  in  Fig.  86,  and  make 
the  vernier  read  zero.  Direct  the  telescope  along 
the  meridian,  clamp  the  instrument,  take  a  back-sight 


FIG.  86. 

upon  the  last  station,  and  proceed  around  the  field  ac- 
cording to  the  process  described  in  §  137.  On  getting 
to  the  last  station  and  looking  back  to  the  first,  the 
fore-sight  should  be  the  same  as  the  back-sight  from 
the  first  station.  Fig.  86  and  Table  I  (page  361)  mutually 
explain  each  other. 

The  angles  in  the  column  headed  fore-sights  are  the 
azimuths  or  angles  which  the  several  courses  make  with 

360 


COMPUTATIONS. 


36: 


the  reference  line,  in  this  case  the  meridian,  and  are 
marked  in  Fig.  86.  If  only  the  area  is  desired,  it  is 
immaterial  whether  the  reference  line  be  a  meridian  or 
not. 

TABLE  I. 
FIELD  NOTES  FOR  TRAVERSE  SURVEYING. 


Sta 

Back-sight. 

Fore-sight. 

Dist. 

Remarks. 

A 

320°  19' 

63°  48' 

Vernier  A    read,    zero 

B 

63°  48' 

123°  2l' 

being     set    on    true 

C 

123°  2l' 

1  80°  00' 

meridian. 

D 

1  80°  00' 

260°  35' 

Azimuths  counted  from 

E 

260°  35' 

320°  19' 

north  toward  east. 

It  makes  little  or  no  difference  which  way  the  sur- 
veyor goes  around  the  field. 

2.  COMPUTING  THE  AREA.  In  computing  the  area  by 
the  common  or  Rittenhouse  method,  there  is  no  check 
upon  the  work  after  the  latitudes  and  departures  have 
been  balanced.  The  principal  object  of  the  method 
explained  below  is  to  overcome  this  defect. 

With  the  following  explanation  of  the  nomenclature, 
the  method  can  readily  be  understood  by  an  inspection 
of  Tables  II  and  III,  pages  362  and  363. 

A^  A^,  A3l  etc.,  are  the  azimuths  of  the  several  sides, 
respectively. 

Clt  C2,  C3,  etc.,  are  the  lengths  of  the  several  sides, 
respectively. 

•*i»  xi->  X^  etc->  are  tne  latitudes  of  the  several  sides, 
respectively. 

y\9  y*>y»>  etc->  are  t^6  departures  of  the  several  sides, 
respectively. 

Xl ,  X^ ,  X3 ,  etc.,  are  the  total  latitudes  of  the  several 
corners;  i.e.,  the  total  distance  which  any  corner  is 
north  or  south  of  some  other  corner  which  is  assumed 
as  an  initial  one  for  the  purposes  of  computation. 


362  AREA    BY    TRAVERSING    WITH    TRANSIT.     [APPEN.  II 


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COMPUTATIONS. 


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364  AREA    BY    TRAVERSING    WITH    TRANSIT.    [APPEN.  II 

Yl9  F2,  F3,  etc.,  are  the  total  departures  of  the  several 
corners. 

2  represents  the  sum  of  the  quantities  to  which  it  is 
prefixed. 

n  is  the  total  number  of  sides. 

T.  Z.,  in  the  heading  of  Table  III,  stands  for  total 
latitude,  and  A.  T.  L.  for  adjacent  total  latitude,  that  is  for 

xlf  Ai  +  jr.,   ^2  +  ^3,   etc. 

A.  D.,  in  Table  III,  represents  the  sums^  -j-jya ,  J2  +^3> 
etc.,  of  Table  II,  which  are  called  adjacent  departures. 

In  computing  the  latitudes  and  departures,  attention 
must  be  given  to  the  signs  of  the  trigonometrical 
functions.  The  latitudes  and  departures  are  to  be 
balanced  as  in  the  method  of  ordinary  land  surveying 
In  applying  this  method,  it  will  be  necessary  to  pre- 
pare a  column  in  which  to  write  the  adjacent  depar- 
tures— yl  +jy2,  y9-\-y3,  etc., — and  another  in  which 
to  write  the  sums  of  the  adjacent  total  latitudes — 
X^-^X^,  Xz-\-X3l  etc.  These  columns,  and  two 
similar  ones  for  the  third  and  fourth  methods,  were 
omitted  from  Table  II  for  convenience  in  printing. 

If  only  two  computations  of  the  area  are  to  be  made 
(two  will  generally  give  a  sufficient  check),  it  is  shorter 
to  use  either  the  first  and  second  method  or  the  third 
and  fourth,  than  to  pair  them  differently.  For  simply  a 
numerical  check,  the  signs  of  the  area  may  be  dis- 
regarded; but  if  the  signs  in  the  table  are  conformed 
to,  the  areas  will  agree  in  sign  as  well  as  in  amount. 

Notice  that  in  the  first  method,  each  partial  area  is 
obtained  by  multiplying  the  total  latitude  of  any  corner 
by  the  sum  of  the  departures  of  the  adjacent  sides,  or 
briefly,  each  partial  area  is  the  product  of  the  total 
latitude  and  the  sum  of  the  adjacent  departures;  and 
note  also  that  in  the  second  method  each  partial  area 
is  the  product  of  the  departure  and  the  sum  of  the 
adjacent  total  latitudes.  Note  further,  that  the  third 


COMPUTATIONS.  365 


and  fourth  methods  are  the  same  as  the  first  and  second 
excepting  the  substitution  of  latitudes  for  departures. 
The  fourth  method  is  the  usual  one  of  double  meridian 
distances,  except  that  the  former  deals  with  the  co- 
ordinates of  the  corners  of  the  field  instead  of  the 
middle  of  the  sides. 

The  example  shown  in  Table  III  is  a  problem  solved 
by  one  of  the  author's  students  in  the  ordinary  class 
work.  For  reasons  not  necessary  to  explain,  the  com- 
putations are  carried  to  an  unusual,  and  ordinarily  an 
indefensible,  number  of  decimal  places. 


APPENDIX   III. 

PROBABLE   ERROR. 

1.  ALL  quantities  determined  by  observation  are  sub- 
ject to  error,  and  any  one  who  deals  with  such  quanti- 
ties should  recognize  the  certainty  of  error  in  his  data. 
This  limitation  is  frequently  disregarded — as,  for 
example,  when,  in  a  report  which  recently  came  under 
the  author's  eye,  the  cost  per  cubic  yard  of  concrete 
was  figured  out  to  the  thousandth  of  a  cent.  Other 
examples  could  be  cited  in  which  the  result,  although 
less  ridiculous,  involved  vastly  greater  consequences. 
The  object  of  this  discussion  is  to  present  some  of 
the  elementary  principles  involved  in  the  comparisons 
of  results  derived  from  observation. 

All  observations  are  liable  to  three  classes  of  error; 
viz.,  mistakes,  constant  errors,  and  accidental  errors, 
(i)  Mistakes  are  errors  due  to  inexperience,  to  lack  of 
care,  to  mental  confusion,  etc.;  as,  for  example,  reading 
28°  instead  of  32°,  or  reading  the  wrong  vernier,  or 
counting  from  the  wrong  end  of  the  tape.  Frequently 
such  errors  may  be  corrected  by  comparison  with  other 
observations.  (2)  Constant  errors  are  those  produced 
by  well  understood  causes — as,  for  example,  the  chain 
may  be  too  long,  or  it  may  be  used  at  a  temperature 
differing  from  that  at  which  it  is  of  standard  length,  or 
there  may  be  phase  in  the  target  sighted  at,  etc.  Such 
errors  can  always  be  eliminated  by  the  application  of 
computed  corrections,  and,  strictly  speaking,  are  not 
errors  at  all.  (3)  Accidental  errors  are  those  still  re- 
maining after  all  evident  mistakes  and  all  constant 

366 


PRINCIPLES.  367 


errors  have  been  eliminated — as,  for  example,  the  errors 
in  leveling  due  to  a  movement  of  the  bubble  after  its 
inspection,  to  an  imperfect  bisection  of  the  target,  to 
an  inclination  of  the  rod,  etc.  Only  the  last  class  of 
errors  will  be  considered  here. 

To  many  persons  it  seems  strange,  and  even  im- 
proper, to  speak  of  the  law  of  irregular  and  unknown 
errors  ;  but  it  is  nevertheless  true  that  even  accidental 
errors  follow  a  rigorous  mathematical  law — the  law  of 
probability. 

The  whole  theory  of  probable  error  depends  upon 
the  three  following  theorems,  derived  from  experience: 

1.  Small  errors  are  more  frequent  than  large  ones. 

2.  Positive  and  negative  errors  are  equally  frequent. 

3.  Very  large  errors  do  not  occur. 

2.  The  probable  error  is  such  a  quantity  that  there 
is  an  even  chance  that  the  real  error  is  greater  or  less 
than  it.  Or,  in  other  words,  if  the  errors  of  a  series  of 
observations  were  arranged  in  the  order  of  their  mag- 
nitude, the  middle  error  in  that  series  would  be  the 
probable  error.  Notice  that  the  probable  error  would 
occur  in  the  middle  of  the  series,  but  would  be  less 
than  the  mean  of  the  errors,  because  small  errors  are 
more  likely  to  occur  than  large  ones.  Or,  again,  the 
probable  error  is  taken  by  mathematicians  as  the  limit 
within  which  it  is  as  likely  as  not  that  the  truth 
will  fall.  Thus,  if  5.45  be  the  mean  of  a  number 
of  determinations,  and  0.20  be  the  probable  error, 
then  the  true  result  is  as  likely  to  lie  between  5.25 
(=  5-45  —  0.20)  and  5.65  (=  5.45  +  0.20)  as  to  lie  out- 
side of  these  limits.  The  probable  error  is  usually 
represented  by  writing  it  after  the  number,  but  preced- 
ing it  by  the  sign  indicating//^  or  minus.  For  exam- 
ple :  5.45  ±  o.  20  indicates  that  5.45  is  the  mean  of  the 
observed  quantity,  and  that  0.20  is  the  probable  error  of 
this  value. 


368  PROBABLE    ERROR. 


3.  When  a  number  of  separate  observations   of  any 
kind  have   been   made   with  equal  care,  the    mean   or 
average  of  them  all  is  the  most  probable  value  of  the 
quantity  sought,  and  is  considered  the  true  value. 
Let     n  —  the  number  of  the  observations. 

d  —  the  difference  between  any  one  observation 
and  the  mean  of  the  observations.  (The 
quantities  represented  by  d  are  generally 
called  residuals.) 

El  =  the  probable  error  of  a  single  observation. 
f  =  the  mean  of  the  errors. 
£m  =  the  probable  error  of  the  mean  of  all  the 

observations. 

q  —  0=6745 — a  constant  determined  by  com- 
putations according  to  the  theory  of 
probability. 

2  =  a  symbol  signifying  sum  of* 
Then 


£l  = 


El  =  0.84537,  approximately.     .     (3) 

Less  labor  is  required  in  applying  equation  (3)  than 
in  using  (i).  If  the  number  of  observations  were  in- 
finite, the  two  would  give  exactly  the  same  results. 

Mathematicians  agree  far  better  as  to  the  form  of  the 
law  of  error,  and  also  as  to  the  method  of  computing 
the  probable  error,  than  they  do  as  to  the  manner  in 
which  the  law  can  be  deduced.  Therefore  no  demon- 
stration will  be  attempted  ;  but  to  prevent  misappre- 
hension it  may  be  well  to  remark  that  the  law  as  stated 
above  has  frequently  been  tested  and  found  to  agree 


EXAMPLES.  369 


very  closely  with  experience.  It  should  be  borne  in 
mind  that  the  method  is  grounded  upon  the  hypothesis 
that  a  large  number  of  observations  have  been  taken. 
However,  when  only  a  limited  number  of  observations 
have  been  made,  the  probable  error,  when  computed 
according  to  the  above  formulas,  is  sufficiently  exact 
for  all  purposes  of  comparison.  It  should  not  be  for- 
gotten that  the  probable  error  can  be  computed  legiti- 
mately only  after  all  constant  errors  have  been  cor- 
rected. Nor  should  it  be  forgotten  that  the  probable 
error  considers  only  errors  that  are  as  likely  to  occur 
on  one  side  as  on  the  other. 

4.  Tables  I  and  II  (pages  370  and  371)  will  illus- 
trate the  method  of  applying  the  preceding  formulas. 
The  observations  were  made  by  leveling  up  the  instru- 
ment, sighting  the  target,  and  then  reading  the  rod. 
The  instrument  was  then  disleveled,  the  target  was 
moved,  and  another  observation  was  made.  The  ob- 
servations as  recorded  were  consecutive,  /'.*.,  no  poor 
ones  were  thrown  away.  The  observations  were  made 
by  one  of  the  author's  students  in  ordinary  class-work, 
and  are  fairly  representative  of  what  an  inexperienced 
but  very  careful  man  can  do  under  favorable  conditions 
as  to  wind,  light,  etc.,  with  an  ordinary  wye  level. 

Incidentally  the  tables  give  some  data  concerning  the 
relative  accuracy  of  two  forms  of  level  targets  (§  268), 
and  also  show  the  effect  of  increasing  the  length  of 
sight.  All  the  observations  were  made  by  the  same 
man  on  the  same  day.  For  the  distances  in  the  table, 
the  error  of  setting  the  target  seems  to  vary  about  as 
the  square  root  of  the  length  of  sight,  but  the  error 
probably  decreases  with  the  distance  for  a  time  and 
then  increases.  Notice  that  the  probable  error  of 
setting  the  quadrant  target  (the  one  generally  used  in 
practice)  at  300  feet  (about  the  ordinary  distance)  was 
0.002  ft.  Under  the  usual  conditions  this  error  would 


370 


PROBABLE    ERROR. 


TABLE    I. 

ERROR  OF  READING  LEVEL  TARGET. 
Rod  100  Feet  from  Instrument. 


QUADRANT  TARGET. 

DIAMOND  TARGET'. 

Reading 
in  feet. 

dm 
thousandths. 

d*. 

Reading 
in  feet. 

dm 

thousandths. 

d*. 

3.169 

0 

o 

3.172 

I 

I 

3.170 

I 

i 

3.174 

I 

I 

3.170 

I 

i 

3-174 

I 

I 

3.170 

I 

i 

3.172 

I 

I 

3.169 

0 

0 

3-173 

0 

0 

3-168 

I 

I 

3.173 

O 

0 

3.I7I 

2 

4 

3-173 

o 

o 

3.168 

I 

i 

3-173 

0 

0 

3.169 

O 

o 

3.175 

2 

4 

3.168 

I 

i 

3-174 

I 

i 

3.169 

0 

0 

3.173 

0 

0 

3.170 

I 

I 

3-174 

I 

I 

3.169 

0.75 

II 

3  173 

O.67 

IO 

mean 

mean 

sum 

mean 

mean 

sum 

Prob.  error  of  single  obs. 

Prob.  error  of  single  obs. 

=  0.67  y  —  =  0.00067  ft. 

=  0.67  y  —  =  0.00064  ft- 

Approx.  prob.  error  of  single  obs. 

Approx.  prob.  error  of  single  obs. 

=  0.84  X  0.75  =  0.00063  ft. 

=  0.84  X  0.67  =  0.00056  ft. 

Prob.  error  of  mean 

Prob.  error  of  mean 

0.67 

=  =£  =  0.00019  ft- 

4/12 

0.64 
=  —  =  =  0.00018  ft. 

4/12 

probably  be  considerably  more  than  this — possibly  two 
or  three  times  as  great. 

5.  To  resume  the  discussion  of  the  general  principles 
of  the  probable  error,  notice  that  according  to  equation 
(2)  the  probable  error  of  the  mean  of  a  number  of 
observations  varies  inversely  as  the  square  root  of 
their  number.  For  example,  if  a  compensating  error 


EXAMPLES. 


371 


TABLE   II. 

ERROR  OF  READING  LEVEL  TARGET. 
Rod  300  Feet  from  Instrument. 


QUADRANT  TARGET. 

DIAMOND  TARGET. 

Reading 
in  feet. 

din 

thousandths. 

d*. 

Reading 
in  feet. 

dm 
thousandths. 

rfa. 

4-843 

5 

25 

4.848 

2 

4 

4-835 

3 

9 

4.85I 

4-836 

2 

4 

4.851 

4-837 

I 

i 

4-852 

4-834 

4 

16 

4.849 

4-834 

4 

16 

4.849 

4.837 

i 

i 

4.851 

4.839 

i 

i 

4-852 

2 

4 

4.838 

0 

o 

4.848 

2 

4 

4-835 

3 

9 

4-850 

0 

0 

4.842 

4 

16 

4.846 

4 

16 

4.841 

3 

9 

4-852 

2 

4 

4-838 

2.6 

107 

4-850 

1.6 

41 

mean 

mean 

sum 

mean 

mean 

sum 

Prob.  error  of  single  obs. 

Prob.  error  of  single  obs. 

=  0.67  JU^-L  —  0.0021  ft. 
r     ii 

=  0.67  A/  $1  =  0.0013  ft. 

Approx.  prob.  error  of  single  obs. 

Approx.  prob.  error  of  single  obs. 

=  0.84  X  2.6  =  0.0022  ft. 

=  0.84  X  1.6  —  0.0013  ft. 

Prob.  error  of  mean 

Prob.  error  of  mean 

21 

36ft. 

13         oo 

x>4ft. 

4/12 

—      o.o 

4/12 

of  o.oi  of  a  foot  per  chain  is  made  in  measuring, 
the  error  in  a  line  100  chains  long  would  be  only 
O.OT  X  1/100  =  0.1  feet,  and  not  o.oi  X  100  =  i.o  foot. 
Again,  if  the  probable  error  of  a  single  reading  of  a 
level  target  is  0.005  feet,  then  the  error  in  determining 
a  difference  of  elevation  by  a  single  setting  of  the 
instrument  is  V~2  X  0.005  feet  =  0.007  feet  nearly,  since 


37 2  PROBABLE    ERROR. 


the  determination  requires  two  settings  of  the  target. 
If  sixteen  such  pairs  of  observations  were  taken  to 
determine  the  difference  in  elevation  of  two  remote 
points,  then  the  probable  error  of  the  difference  of  level 
of  the  extreme  points  would  be  0.007  X  Vi6  =  0.028 
feet. 

The  following  illustrations  show  slightly  different 
forms  of  the  above  conclusion.  If  a  represents  the  mean 
of  a  measured  distance  having  a  probable  error  x,  and 
b  represents  another  distance  having  a  probable  error 
y,  then  the  probable  error  of  the  sum  of  these  distances 
is  W  -|-  y.  Or,  stating  this  algebraically  we  have 

(a  ±  x)  +  (b  ±y)  =  a  +  b  ±  Vx*  +/.  .     .     (^ 

Again,  if  c  ±  x  and  d  ±  y  represent  the  two  sides  of  a 
rectangle,  the  area  will  be  represented  by 


If  z  represents   the  probable  error  per  unit   (say  per 
chain),  then  the  area  is  represented  by 


cd±  zcd(c  +  d) (6) 

To  apply  the  last  formula,  assume  that  a  lot  20  X  100  ft. 
is  laid  out  with  a  chain  with  such  a  degree  of  accuracy 
that  the  probable  error  per  chain  is  o.oi  ft.,  then 


cd ±  z  Vcd(c  -{-  d)  —  2,000  ±  o.oi  ^2,000  (20  -J-  100) 
=  (2,000  ±  4.90)  sq.  ft. 

There  are  various  ways  in  which  the  preceding  prin- 
ciples may  be  made  use  of  in  practical  work,  some  of 
which  will  readily  suggest  themselves,  but  it  is  not  wise 
to  discuss  them  here, 


APPENDIX  IV. 

PROBLEMS  IN  TESTING,  ADJUSTING,  AND 

USING    ENGINEERS'    SURVEYING 

INSTRUMENTS. 

THE  following  problems  are  solved  by  the  author's 
students  in  connection  with  the  study  of  the  preceding 
text.  Some  of  them  are  solved  several  times — with  dif- 
ferent instruments  and  under  different  conditions  as 
to  distance,  weather,  experience,  etc.  A  report  of  each 
problem  is  made  upon  a  sheet  of  standard  co-ordinate 
paper. 

PROB.  1.     ERROR  OF  MEASURING  WITH  A  STEEL  TAPE. 

1.  Measure  a  distance  of  about   1,000  feet  with  a  steel 
tape.     Do  not  use  a  spring  balance  or  a  thermometer, 
but  take  every  other  precaution  to  secure  extreme  ac- 
curacy.    Make  at  least  four  measurements,  each  man 
being  fore  chain-man  at  least  once  in  each   direction. 
Compare  the  tape  with   the   standard,  both  before  and 
after  taking  the  measurement.     Report  the  true  horizon- 
tal distance  between  the  ends  of  the  line. 

2.  Compute  the  probable  error  of  a  single  measure- 
ment and  also  of  the  mean. 

3.  Estimate  the  values  of  the  errors  due  to  the  several 
sources   mentioned  in  §  19,  i.e.,  assign   values   to  a,  b,  c, 
etc.,  in  equation  (2).  page  26;  and  compare  the  estimated 
probable  error  of  a  single  measurement  with  the  cor- 
responding observed  probable  error. 

373 


374  PROBLEMS. 


4.  Level  the  line  and  compute  the  correction  for 
inclination.  How  does  the  estimated  quantity  agree 
with  the  true  value  ? 

PROB.  2.     ERROR  OF  MEASURING  WITH  TWO  STEEL  RULES. 

i.  To  determine  the  degree  of  accuracy  with  which  dis- 
tances can  be  laid  off  by  means  of  short  rules,  use  two 
2-foot  steel  rules  and  lay  off  50  feet  on  the  floor.  The 
rules  may  be  used  for  either  end  or  line  measurement  ; 
/.*.,  the  rules  may  be  placed  end  to  end,  or  they  may  be 
lapped  and  the  coincidence  of  lines  observed.  Make 
three  measurements  of  the  distance  by  each  method. 

Report  which  method  is  considered  the  better,  and 
state  the  reasons  therefor. 

PROB.  3.     ANGLES  WITH  A  TAPE. 

1.  With  a  steel  tape,  measure  four  angles  around  a 
point  such  that  their  sum  =  360°,  no  two  of  the  angles 
being  of  the  same  size. 

2.  Measure  the  three  angles  of  a  triangle  having  sides 
about  250  to  300  feet  long. 

3.  Report  the  sums  in   each  case,  and   also  state  the 
radius  used,  the  form  of  target  (flag  pole  or  chaining 
pin),  weather,  etc. 

PROB.  4.     ADJUSTMENT  OF  THE  MAGNETIC  COMPASS. 

1.  Test  to  see  whether  the  center  of  graduation  is  in 
the  line  of  sight. 

2.  Test  to  see  if  the  zero  of  the  vernier  is  in  the  line 
of  sight. 

3.  Adjust  the  levels  perpendicular  to  the  vertical  axis. 

4.  Place  the  sights  (a)  in  the  same  plane,  and  (b)  per- 
pendicular to  the  plate. 

5.  Place  the  point  of  suspension  of  the  needle  (a)  in 


MAGNETIC    COMPASS.  375 

the  vertical  plane  of  the  ends,  and  (fr)  in  the  horizontal 
plane  of  the  ends. 

6.  Sharpen  the  pivot  and  place  it  in  the  center  of  the 
graduation. 

7.  If  the  instrument  is  provided  with  a  means  of  set- 
ting off  a  perpendicular,  test  it. 

8.  Test  the  magnetism  of  the  needle. 


PROB.  5.     ERROR  OF   SIGHTING  A  MAGNETIC  COMPASS. 

1.  Set  up  the  magnetic  compass  and  clamp  the  verti- 
cal axis.     At   TOO  feet  from   the  instrument,  set  a  flag 
pole  in  the  line  of  the  slits,  and  mark  its  position  on  a 
board    laid    on    the   ground   for   that    purpose.      Then 
slightly  change  the  position  of  the  flag  pole,  and  line  it 
in    again.     Make   at   least    ten  sightings   on    the    pole. 
Measure   the  position  of  the  marks  from  one  end  of  the 
board,  and  compute  the  linear  distance  corresponding  to 
the  probable  error  of  sighting.     Reduce  the  linear  error 
to  angular  error. 

2.  Repeat  the  above  process  at  300  and  also  at  600 
feet. 

PROB.  6.     ANGLES  WITH  A  MAGNETIC  COMPASS. 

i.  With  a  compass  measure  four  angles  around  a 
point,  such  that  their  sum  —  360°.  Make  the  observa- 
tions as  follows  :  Sight  upon  the  first  line  and  read  the 
needle;  sight  the  second  line  and  read  again.  The  dif- 
ference of  these  readings  is  the  first  angle.  Move  the 
vernier  a  little  to  eliminate  personal  bias,  and  turn  the 
compass  slightly  to  eliminate  any  sticking  of  the  needle; 
then  measure  the  second  angle  as  before.  Do  similarly 
for  the  other  angles. 

Try  to  have  the  conditions  as  to  length  of  sight,  tar- 
gets, etc.,  as  nearly  as  possible  like  those  of  Prob.  3. 


376  PROBLEMS. 


2.  Measure  the  three  angles  of  a  triangle  having  sides 
250  to  300  feet  long. 

3.  Report  the  sums  in   each  case;  and   also  state  the 
form  of  target,  length  of  sight,  and  condition  of  wind, 
sun,  etc. 

PROB.  7.     ELIMINATION   OF   LOCAL  ATTRACTION. 

1.  With  a  magnetic  compass  observe  the  back-sights 
and  fore-sights  of  the  several  sides  of  a  field  of  at  least 
five   sides.      Read    the   bearings    to   the   nearest    five 
minutes. 

2.  Correct  the  field  notes  to  eliminate  local  attraction, 
and  to  determine  the  angular  error  of  closure. 

PROB.  8.     TRUE   MAGNETIC  BEARINGS. 

1.  Determine  the  true  bearings  of  the  several  sides 
of  a  field  with  a  magnetic  compass  by  connecting  one 
corner  of  the  field  with  a  true  meridian.     See  §  n,  Ap- 
pendix II. 

2.  Find  the  true  bearings  and  also  the  angular  error 
of  closure. 

PROB.  9.     TESTING  A  TELESCOPE. 

1.  Test  (a)  spherical  aberration,  (b)  chromatic  aberra- 
tion, (c)  defining  power,  and  (d)  flatness  of  field. 

2.  Measure  the  magnifying  power.     Use  two  methods 
and  make  three  determinations  by  each. 

3.  Measure  the  angular  width  of  the  field  of  view. 

4.  Compare  two   telescopes  (a)  for  illumination,  and 
(b)  for  definition.     Are  the  illuminating  powers  of  the 
two  telescopes  in  the  direct  ratio  of  the  areas  of  real 
apertures  and  in  the  inverse  ratio  of  the  squares  of  the 
magnifying  powers  ? 


TRANSlf.  377 


PROB.  10.     STRETCHING  SPIDER  WEBS. 

1.  Attach  three  spider  webs  to  a  diaphragm,  parallel 
to  each  other  and  -g*¥th  of  an  inch  apart.     Be  sure  that 
all  these  are  in  the  same  plane. 

2.  Fasten  a  fourth  web  perpendicular  to  the  others 
and  in  the  same  plane. 

3.  After  the  webs   are  dry,  breathe   upon    them    or 
hold   them  in  a  gentle  current  of  steam.     If  they  be- 
come slack,  they  were  not  sufficiently  stretched  before 
being  fastened. 

PROB.  11.     MAKING  A  VERNIER. 

i.  Graduate  a  scale  of  equal  parts  on  a  piece  of  Bris- 
tol board,  using  a  right-line  pen  and  India  ink,  and 
make  a  vernier  to  read  the  scale.  The  unit  of  the  scale 
and  vernier  will  be  assigned.  The  accuracy  of  the 
graduation  is  the  most  important  feature  of  this 
problem. 

PROB.  12.     ADJUSTMENT  OF  A  TRANSIT. 

1.  Test  the  eccentricity  of  the  graduation. 

2.  Adjust  the  plate  levels   perpendicular  to  the  ver- 
tical axis. 

3.  Adjust  the  line  of  collimation  perpendicular  to  the 
horizontal  axis. 

4.  Adjust   the  horizontal  axis   perpendicular   to  the 
vertical  axis. 

5.  Test   the    motion    of   the    objective    in    a   vertical 
plane. 

6.  Adjust  the  level  under  the  telescope   (if  there  is 
one)  parallel  to  the  line  of  collimation,  by  the  two-peg 
method. 

7.  Adjust  the  zero  of  the  vertical  circle  to  read  zero 
when  the  line  of  sight  is  horizontal. 


PROBLEMS. 


PROB.  13.     ERROR  OF  SIGHTING  A  TRANSIT. 

1.  Set  up  the  transit  and  clamp  the  vertical  axis.     At 
100  feet  from  the  instrument  set  a  flag  pole  in  the  line 
of  sight  of  the  telescope,  and  mark  the  position  of  the 
pole  on  a  board  laid  on   the  ground  for  that  purpose. 
Slightly  change  the  position  of  the  flag  pole,  and  line 
it  in  again.     Make  at  least  ten  sightings  on  the  pole. 
Measure  the  position  of  the  marks  on  the  board,  from 
one  end  of  it,  and  compute  the  linear  distance  corre- 
sponding to  the  probable  error  of  sighting.       Reduce 
the  linear  error  to  angular  error. 

2.  Repeat  the  above  process  at  300  and  also  at  600 
feet. 

PROS.  14.     ANGLES  WITH  TRANSIT. 

1.  With  a  transit  and  the  other  conditions  about  as  in 
Probs.  3  and  6,  measure  the  four  angles  around  a  point. 
Make  the  observations  as  follows:  Sight  upon  one  tar- 
get and  read  the  vernier,  then  turn  to  the  second  target 
and  read  again.     Next  turn  the  instrument  a  trifle,  with 
the  lower  movement,  to  eliminate  personal  bias,  and 
sight   upon   the  second  target  and  read   the   vernier ; 
then   turn  to  the  third   target  and  read  again.     In  a 
similar  manner  measure  all  four  of  the  angles. 

2.  Measure  the  three  angles  of  a  triangle.     Measure 
both  the  interior  and  exterior  angle  at  each  station  as  a 
check. 

3.  Report  the  sum  of  the  four  and  also  of  the  three 
angles,  and  state  the  length  of  sights,  the  kind  of  tar- 
gets, the  least  count  of  the  vernier,  the  condition  of  the 
weather,  etc. 


TRANSIT.  370 


PROB.  15.     ANGLES  WITH  A  TRANSIT  BY  REPETITION. 

1.  With  the  conditions  as  nearly  as  possible  the  same 
as    in    Prob.    14,    measure    the    three    angles   of   a    tri- 
angle by   repetition.     Repeat  each    angle  three  times. 
Measure   both   the   interior   and   exterior   angles   as  a 
check. 

2.  Report  the  sum  of  the  three  angles,  and  the  details 
of  the  work. 

PROB.  16.     TRAVERSING  WITH   A  TRANSIT. 

i.  With  a  transit,  determine  the  azimuths  of  the 
several  sides  of  a  field,  using  one  of  the  sides  as  the  ref- 
erence line. 

What  will  be  the  effect  of  an  error  of  collimation  ? 

PROB.  17.     TRAVERSING  WITH  A  TRANSIT. 

i.  With  a  transit,  determine  the  azimuths  of  the  several 
sides  of  a  field,  using  a  true  meridian  as  the  reference 
line. 

PROB.  18.     AREA  WITH  A  TRANSIT. 

1.  Determine  the  area  of  a  field  having  at  least  five 
sides,  using  a  transit  and  a  roo-foot  steel  tape. 

2.  Compute  the  area  by  two  methods  (see  Tables  II 
and  III,  Appendix  III). 

3.  Report  the  field  notes  and  the  plat  upon  one  sheet 
of  regulation  co-ordinate  paper,  and  the  computations 
upon   another.     State   the  area  in  acres  and   decimals 
thereof. 

4.  Try  to  secure  uniform  accuracy  throughout.     How 
many  places  of  logarithms  should  be  used  ?     As  worked, 
what  is  the  weakest  link  ? 


380  PROBLEMS. 


PROB.  19.     MERIDIAN  WITH  SOLAR  TRANSIT. 

1.  Determine  the  direction  of  the  zero  line  of  the  sur- 
veying spiral,  with  a  solar  transit.     Make  three  obser- 
vations in  the  forenoon. 

2.  Make  an  equal  number  of  observations  in  the  af- 
ternoon. 

3.  What  conclusions  can  be  drawn  from  a  comparison 
of   the   forenoon    and    afternoon    observations?     What 
conclusion   can    be    drawn    from   a  comparison   of    the 
mean  of  all  the  observations  and  the  true  direction  of 
the  meridian  ? 

PROB.  20.     TESTING   A  HOME-MADE  PLANE  TABLE. 

1.  Test  the  straightness  of  the  edge  of  the  alidade. 

2.  Test  the  sights  to  see  (a)  if  they  are  in  a  plane,  (b) 
if  they  are  perpendicular  to  the  board,  and  (c)  if  their 
plane  passes  through  the  edge  of  the  alidade. 

3.  Mark  the  point  at  which  the  centre  of  the  bubble 
should  stand  when  the  top  of  the  board  is  level. 

PROB.  21.     ANGLES  WITH   A  PLANE  TABLE. 

1.  Lay  down  four  angles  on  paper,  such  that  their  sum 
=  360°,  no  two  being  of  the  same  size.     Determine  the 
value  of  each   by  measuring   the  chords  with  a  scale. 
The  difference  between  the  sum  of  the  observed  values 
and  360°  shows  the  error  of  scaling  off.     Compute  the 
probable  error  of  scaling  off  a  single  angle.     Compare 
this  error  with  that  of  paragraph  i,  Prob.  3. 

2.  With  the  conditions  as  nearly  as  possible  like  those 
in  Probs.  3  and  6,  measure  the  three  angles  of  a  triangle. 
Compute  the  probable  error  of  a  single  angle.     Com- 
pare this  error  with  those  of  Probs.  3  and  6. 

3.  Report  the  results,  and  state  the  length  of  radius, 
kind  of  scale  used,  etc.,  and  also  the  form  of  target,  the 
condition  of  the  weather,  etc. 


PLANE   TABLE.  3$] 


PROB.  22.     ANGLES  WITH  A   PLANE   TABLE  BY   REPETI- 
TION. 

1.  Set  out  on  the  ground  an  approximately  equi-angu- 
lar  triangle,  having  the   length  of  sides,  targets,  etc., 
about  as  in  Prob.  3,  6,  and  14. 

2.  Measure  each  angle  with  the  plane  table  by  "re- 
peating "  it  six  times.     Scale  off  the  difference  between 
six  times  the  angle  and  360°.     Make  this  measurement 
with  two  different  radii  or  two  different  scales,  or  both, 
to  reduce  the  error  of  measuring  this  angle.     Subtract 
this  difference  from  360°,  and  divide  the  result  by  six  to 
get  the  observed  value  of  the  angle  of  the  triangle. 

3.  Report  the  sum  of  the  three  angles  of  the  triangle, 
the  length  of  radius  and  kind  of  scale  used  in  scaling 
off  the  angle,  the  length  of  sight,  the  kind  of  targets, 
the  weather,  etc. 

PROB.  23.    AREA  WITH  A  PLANE  TABLE  BY  RADIATION. 

1.  Find  the  area  of  a  field  by  the  method  of  radia- 
tion. 

2.  Present  a  reduced  plat  of  the  field,  and  state  the 
scale  of  the  plat  shown  and  also  the  scale  of  that  from 
which  the  area  was  determined.     Show  the  directions 
of    the  cardinal    points.     State  the  area  in  acres  and 
decimals. 

PROB.  24.    AREA  WITH   A  PLANE  TABLE  BY  TRAVERS- 
ING. 

1.  Find  the  area  of  a  field  by  traversing. 

2.  Make  a  report  as  in  paragraph  2  of  Prob.  23. 

PROB.  25.  AREA  WITH  A  PLANE   TABLE   BY  RADIO-PRO- 
GRESSION. 

1.  Find  the  area  of  a  field  by  radio-progression. 

2.  Make  a  report  as  in  paragraph  2  of  Prob.  23. 


PROBLEMS. 


PROB.  26.     THREE-POINT  PROBLEM. 

1.  Knowing  the  lengths  of  the  three  sides  of  a  tri- 
angle  located  upon  the  ground,  and  having  the  position 
of  a  fourth  point  given  upon  the  ground,  set  the  plane 
table  over  the  latter  and  determine  its  position  on  the 
map  by  a  mechanical  solution. 

2.  Check  the  above  work  by  a  graphical  solution. 

3.  Present  a  reduced  plat  of  the  work  of  the  second 
method,  giving  the  scale  of  the  plat  shown  and  stating 
that  of  the  plat  used  in  the  field.     State  the  difference 
between  the  results  by  the  two  methods, 

PROB.  27.     TWO-POINT  PROBLEM. 

1.  Knowing  the  distance  between  two  "  inaccessible  " 
points  and  having  a  third  point  given  upon  the  ground, 
determine  the  position  of  the  latter  upon  the  paper  by 
solving  the  two-point  problem  with  the  plane  table. 

2.  Measure  the  distance  from  each  of  the  "inacces- 
sible" points  to  the  point  over  which  the  instrument  is 
set,  and  plat  the  true  position  of  this  point. 

3.  Present  a  reduced  plat  of   the  work,  stating   the 
scales  used  in  the  field  and  in  making  the  reduced  plat. 

PROB.  28.     STADIA  CONSTANTS. 

1.  Establish  eight  or  ten  points  approximately  in  line 
at  irregular  intervals  on  nearly  level  ground. 

2.  Set  the  instrument  near  the  end  of  the  line,  and 
with  the  stadia  determine  the  distance  from  the  instru- 
ment to  each  point. 

3.  With  a  chain  or  steel  tape  measure  the  distance 
from  the  instrument  to  each  point.     Measure  with  the 
stadia,  first  to  avoid  the  possibility  of  any  personal  bias 
in  reading  the  stadia  rod. 

4.  Measure  c  and  /. 


STADIA.  383 


5.  Compute  k  in   the   formula  D  =  ks-}-c-\-f,  for 
each  distance. 

6.  Find  the  mean  value  of  k,  and  compute  the  prob- 
able error  (in  linear  units  on  the  ground)  of  a  single 
observation,  and  also  the  probable  error  of  the  mean. 

PROB.  29.     HORIZONTAL  DISTANCES  WITH  THE  STADIA. 

1.  Locate  a  number  of  points  approximately  in  line 
at  irregular  intervals. 

2.  Use  the  same  instrument  and   rod  as  in  Prob.  28, 
and  determine  the  horizontal  distances  with  the  stadia. 

3.  Rod-man  and  instrument-man  change  places,  and 
re-determine  the  distances. 

4.  Measure  the  distances  with  a  steel  tape. 

5.  Report   the   two    series  of  stadia-determined   dis- 
tances, and    also    the  true    distances.     Does    the    error 
of  observation  vary  with  the  distance  ?     How  ?    Why  ? 

PROB.  30.     VERTICAL  DISTANCES  WITH  THE  STADIA. 

1.  Use    the   instrument   and  rod    employed  in  Prob. 
28,  and  determine  the  elevation  of  each  point  of  the  sur- 
veying spiral,  with  reference  to  the  central  one. 

2.  Rod-man  and  instrument-man  change  places,  and 
re-determine  the  elevations. 

3.  Report  the  two  series  of  results  in  tabular  form. 

PROB.  31.     ERROR  OF  SETTING  A  LEVEL  TARGET. 

1.  Take  at  least  ten  readings  upon  a  rod  at  100  feet. 
Preserve  at  least  ten  consecutive  results. 

2.  Take  ten  or  more  readings  at  300  feet.     Preserve 
ten  consecutive  results. 

3.  Compute  the  probable  error  for  a  single  reading 
for  each  distance. 

4.  In   the  report  state  the   magnifying  power  of  the 
telescope,  the  radius  of  curvature  of  the  level  vial,  ancj 
the  kind  of  weathei, 


384  PROBLEMS. 


PROB.  32.     ADJUSTMENT  OF  THE  WYE  LEVEL. 

1.  Find  (a)  the  radius  of  curvature  of  the  level  vial, 
(J>)  the  value  in  arc  of  one  division  of  the  scale,  and  (c) 
the  value  in  arc  of  one  inch  of  the  scale. 

2.  Adjust  the  line  of  the  bottoms  of  the  rings  per- 
pendicular to  the  vertical  axis. 

3.  Adjust  the  level  tube  parallel  to  the  bottoms  of 
the  rings. 

4.  Make  the  line  of  collimation  to  coincide  with  the 
axis  of  the  rings.    In  making  this  adjustment  use  a  r.ear 
point,  and  then  test  the  adjustment  for  a  remote  point. 
If  it  is  in  adjustment  for  the  latter,  the  telescope  slidfi  is 
correct  in  all  particulars. 

5.  Test   the    accuracy  ot    the   adjustments   and   the 
equality  of  the  rings,  by  a  check  level. 

PROB.  33.     ADJUSTMENT  OF  THE  DUMPY  LEVEL. 

1.  Find  (a)  the  radius  of  the  curvature  of  the  level 
vial,  (b)  the  value  in  arc  of  one  division  of  the  scale,  and 
(<c)  the  value  in  arc  of  one  inch  of  the  scale. 

2.  Adjust  the  level  tube  perpendicular  to  the  vertical 
axis. 

3.  Adjust  the  line  of  sight  parallel  to  the  level  tube, 
by  the  two-peg  method. 

4.  Repeat  the  adjustment  for  a  check. 

PROB.  34.     DIFFERENTIAL  LEVELING. 

1.  Level  a  circuit  of  about  a  half  mile,  returning  to 
the  point  of  beginning. 

2.  State  the  error  per  mile,  assuming  it  to  vary  as  the 
square  root  of  the  distance.     State  also  the  time  given 
to  the  field  work, 


LEVELING.  385 


PROB.  35.     PROFILE  LEVELING. 

1.  Determine   the    relative   elevations  of  the  several 
points  of  the  surveying  spiral  and  close  upon  the  point 
of  beginning. 

2.  Draw  a  profile  on  profile  paper,  assuming  the  dis- 
tance between  successive  stations  to  be  100  feet. 

3.  Submit  the  original  field  notes  with  the  profile. 

PROB.  36.     PRECISE  LEVELING. 

1.  Level  a  circuit  of  about  a  half  mile,  returning  to 
the  point  of  departure.     Use  the  method  of  double  lev- 
eling with  one  rod — see  III,  Fig.  78,  page  268. 

2.  State  the  error  per  mile,  assuming  it  to  vary  as  the 
square  root  of  the  distance.     State  also  the  time  given 
to  field  work. 

PROB.  37.     ERROR  OF  READING  MERCURIAL  BAROMETER. 

1.  Read  the  attached  thermometer.     Try  to  have  the 
conditions  such  that  the  temperature  of  the  barometer 
will  not  change  during  the  observations. 

2.  Read  the  barometer,  and  preserve  at  least  ten  con- 
secutive results.     Alter  the  vernier  and  also  the  adjust- 
ment of  the  mercury  and  the  ivory  point,  after  each  ob- 
servation, to  eliminate  unconscious  bias. 

3.  Read  the  attached  thermometer  again,  as  a  check 
against  a  change  of  temperature  of  the  instrument. 

4.  Compute  the  probable  error  of  a  single  reading. 

PROB.  38.     LEVELING   WITH   A   MERCURIAL   BAROMETER. 

i.  Determine  the  elevation  of  the  clock  tower  above 
the  observatory,  by  reading  the  barometer  and  the  at- 
tached and  detached  thermometers  at  each  place.  Take 
three  readings  at  the  observatory,  ihree  in  the  tower, 
and  then  three  more  in  the  observatory. 

3.  Reduce  all  the  readings  to  32°  F.     What  do  the 


386  PROBLEMS. 


three  readings  at  each  point  show  as  to  the  accuracy  of 
barometric  leveling  ?  What  do  the  means  of  the  two 
series  at  the  observatory  show  as  to  gradient  error  ? 

3.  Compute  the  difference  of   elevation   by  at  least 
three  standard  formulas,  using  the  mean  of  the  six  read- 
ings at  the    observatory  for   the  reading  at  the  lower 
station,   and   the  mean   of  those  in   the  tower  for  the 
reading  at  the  upper  station. 

4.  After  having  performed  the  above  operations,  ask 
the   instructor  for  the   true  difference  of  level.     What 
does  the  difference  between  the  true  and  the  observed 
result  show  as  to  the  accuracy  of  barometric  leveling? 
Are  the  conditions  of  this  problem  favorable  or  unfavor- 
able as  a  test  of  the  accuracy  of  barometric  leveling  ? 

PROB.  39.     LEVELING  WITH   A   MERCURIAL   BAROMETER. 

Repeat  Prob.  38,  using  two  points  having  a  consider- 
able horizontal  distance  between  them. 

PROB.  40.     LEVELING  WITH  AN  ANEROID  BAROMETER. 

1.  Determine  the  difference  of  elevation  between  the 
floor  of  the   observatory  and  each  floor  of  University 
Hall,  and  also  that  of  the  clock  tower.    Read  the  aneroid 
— both  the  scale  of  inches  and  the  scale  of  elevations — 
and  the  detached  thermometer  at  the  observatory,  and 
also  at  each  floor  on  the  way  up,  and  repeat  at  each  floor 
on  the  way  down  and  at  the  observatory. 

2.  Compute  the  elevation  of  each  floor  above  the  ob- 
servatory, for  each  of  the  two  readings,  by  at  least  two 
formulas. 

3.  Compare  the  computed  differences  with  the  differ- 
ences obtained  from  the  scale  of  elevations.     What  do 
the  differences  show? 

4.  Are  the  conditions  of  this  problem  favorable  or  un- 
favorable as  a  test  of  the  accuracy  of  the  aneroid  ? 


INDEX. 


ABE— BAR 

Aberration,  chromatic,  80 

spherical.  80 

of  sphericity,  74 
Adie's  telemeter.  217 
Adjustments,  general  principles,  43 

see  the  instrument  in  question. 
Alidade,  denned,  146 

construction,  146 
home-made,  151 
Altitude  scale  of  aneroids,  313 
Aneroid  barometer,  altitude  scale,  343 

common.  301 

formula  for,  341 

Goldschmid,  306 

scale  of  inches,  303 

Vidi,  301 

Aperture  of  objective.  82 
Areas,  accuracy  of,  with  chain,  53 
with  compass,  52 
with  plane  table,  169 
with  transit,  127 

Babinet's  barometric  formula,  340 
Bailey's  barometric  formula,  341 
Balancing  latitudes  and  departures,  53 
Barometer,  aneroid,  301 
common,  301 
defects,  303 
formula  for,  341 
merits,  305 
reading.  305 
scale  of  elevations,  343 
temperature  correction  for,  303 
Goldschmid,  306 
Vidi,  301 

correct  ion  for  temperature  of,  334 
forms.  292 
mercurial,  293 
cleaning,  294 
filling,  296 
reading,  299 
transporting,  300 
Vidi.  301 

Barometric  co-efficient,  333 
Barometric  formulas.  327 
air,  temperature  of,  335 
altitude  term.  336 
assumptions,  327 
Babinet's,  340 
Bailey's,  341 
constants,  333 
Ferrel's,  345 


BAR— CHA 

Barometric  formula,  fundamental  rela- 
tions, 329 
altitude  term,  336 
constants,  333 
humidity,  336 
latitude,  335 
temperature  of  air,  335 
temperature  of  barometer,  334,  303 
Gilbert's  formula,  346 
humidity  term,  336 
instrument,  temperature  of,  334 
Laplace's  formula,  338 
latitude  term,  335 
Plantamour's  formula,  342 
temperature  of  air,  335 
temperature  of  instrument,  334 
typical  formulas,  338 
Weilenmann's  formula,  346 
Williamson's  formula.  343 
Barometric  gradient,  308 
annual,  311 
diurnal,  310 
non-periodic,  312 
permanent,  312 
Barometric  leveling,  292 
limits  of  precision,  319 
methods  of  observing,  321 
Gilbert's,  326 

observations  at  selected  times,  322 
Plantamour's,  325 
Ruhlmann's,  326 
simultaneous  observations,  322 
single  observations,  321 
Whitney's,  325 
Williamson's.  323 
sources  of  error,  308 
gradient,  308 
annual.  311 
diurnal,  310 
non-periodic,  312 
permanent,  312 
humidity,  315 
instrument,  316 
observation,  317 
temperature  of  air,  313 
wind,  318 

Benches  in  leveling,  290 
Boston  leveling  rod,  228 
British  leveling  rod,  234 

Chain,  construction,  3 
correction,  12 

387 


388 


INDEX. 


CHA— GOL 

Chain,  engineer's,  4 
Gunter's,  3 
merits  and  defects,  4 
testing,  12 
tying  up,  4 
using,  14 
Chaining,  14 
limits  of  precision,  27 
on  slope,  16 
sources  of  error,  21 
Chromatic  aberration,  80 
Clamp,  96 

Clarke's  telemeter,  217 
Compass,  magnetic,  38 
adjustments,  43 
care  of,  46 
construction,  37 
needle,  41 
adjustment,  45 
test,  41 
tests,  41 
using,  46 

limits  of  precision,  52 
local  attraction,  349 
practical  hints,  47 
sources  of  error,  48 
solar,  57 
adjustments,  61 
construction,  58 
defects,  62 
history,  63 
merits,  62 
principle  of,  57 
Corner  target  for  leveling,  272 
Cross  hairs,  material  of,  78 

stretching,  79 
Curvature,  correction  for,  280 

Declinator,  defined,  148 
Defining  power  of  telescope,  80 
Departures   and  latitudes,  balancing 
of,  53 

level,  adjustments,  247 
collimation,  247 
level,  248 
wyes,  248 

Eckhold's  telemeter,  216 

Error,  probable — see  Probable  error. 

Errors,  compensating  vs.  cumulative, 

19 
Eye-piece,  74 

Ferrel's  barometric  formula,  345 
Foot-plate,  described,  255 
Foot  screws.  33 
Francis  leveling  rod,  234 

Galilean  telescope,  73 

Gautier's  telemeter,  217 

Geodesic  leveling,  267 
field  routine,  270 
methods,  268 
record,  270 
river  crossings,  288 

Gilbert's  barometric  formula,  346 

Gilbert's  method  of  barometric  level- 
ing, 326 

Goldschmid  aneroid,  306 


GBA— LEV 

Gradient,  barometric,  308 
annual,  311 
diurnal,  310 
non-periodic,  312 
permanent,  312 
Gradienter,  construction.  100 
defined, 209 
theory. 209 
as  a  level,  209 
as  a  telemeter,  209 
constant  intercept,  213 
variable  intercept,  210 
vs.  stadia,  214 

vertical  circle  as  agradienter,  215 
Gunter's  chain,  3 
Gurley's  plane  table,  149 

Hints,  practical — see   the    particular 

instrument. 
Huyghen's  eye-piece,  74 

Johnson's  plane  table,  150 
Kellner's  eye-piece,  77 

Laplace's  barometric  formula,  338 
Latitudes  and  departures,  balancing, 

Level,  care  of,  291 
dumpy,  adjustments,  247 
collimation,  247 
level,  248 
wye,  218 

notes,  differential  leveling,  252 
precise  leveling,  267 
profile  leveling,  256 
height  of  instrument,  259,  262 
differences,  262 
peg,  described,  255 
precise,  adjustment,  249 

tests,  249 
rod— see  Rod. 

test  by  two-peg  method,  246 
tests,  236 
bubble  tube,  236 

magnification  vs.  sensitiveness,238 
using,  252— see  also  Leveling, 
wye,  adjusting,  241 
collimation,  242 
cross  hairs,  242 
level  tube,  241 
test  level,  246 
Leveling,  barometric— see  Barometric 

Leveling, 
benches,  290 
deflection  of  plumb,  271 
differential,  252 
field  routine,  253 
record,  255 

instruments,  adjustment,  dumpy ,247 
geodesic,  249 
wye,  241 

construction,  218 
classified,  219 
dumpy,  223 
geodesic.  224 
precise,  224 
wye,  220 
testing,  236 


INDEX. 


339 


LEV— NOT 

Leveling,  testing  bubble  tube,  236 

sensitiveness,  237 
magnification  vs.  sensitiveness, 

238 

limits  of  precision,  281 
reciprocal,  288 
river  crossings,  288 
sight,  length  of,  287 
sights,  equality  of,  28? 
speed,  284 
practical  hints,  285 
precise,  267 
field  routine,  269 
methods,  267 
record,  270 
river  crossings,  288 
profile,  256 
field  routine,  257 
record,  258 
by  differences,  262 
by  height  of  instrument,  259 
improved,  262 
modified,  259 
reciprocal,  288 

precision  of,  289 
river  crossing,  288 
screws,  33 
sights,  length  of,  287 

equality  of,  287 
sources  of  error,  271 
computing,  278 
curvature,  278 
instrumental,  272 
observational,  273 
personal,  276 
recording,  278 
refraction,  278 
rod,  272 

settling  of  turning  points,  273 
speed,  284 
waving  rod,  254 

Limits  of  precision — see   the  instru- 
ment in  question. 
Local  attraction,  land  surveys,  356 
mine  surveys,  349 

Magnetic  compass,  38 
adjustments,  43 
cai-e  of,  46 
construction,  37 
needle,  adjustment,  45 

test,  41 
tests,  41 
using,  46 

limits  of  precision,  52 
local  attraction,  349 
practical  hints,  47 
sources  of  error,  48 
Magnetic  declination,  varies  with  the 
instrument,  50 
variation,  elimination  of,  349 
Micrometer,  71 
Mine  surveys,  local  attraction  in,  349 

Needle,  magnetic — see  Magnetic  Corn- 


New  York  leveling  rod,  227 
Notes,  method   of   keeping — see    the 
particular  instrument. 


OBJ— HOD 

Objective,  aperture  of,  82 
Objective  slide,  dumpy  level,  240 

transit,  99 

wye  level,  239 

Parallax  in  telescope,  88 
Philadelphia  leveling  rod,  228 
Plane  table,  adjustments,  153 
construction,  146 
complete,  146 
Gurley's,  149 
home-made,  151 
Johnson's,  150 
light,  149 

limits  of  precision,  169 
yractical  hints,  170 
sources  of  error,  168 
tests,  153 

using,  method  of,  155 
intersection,  163 
progression,  157 
radiation,  156 
radio-progression,  159 
three-point  problem,  165 
Plane  table,  traversing,  157 
two-point  problem,  166 
Plantamour's  barometric  formula,  S42 
Plan  I  amour's  method  of  barometric 

leveling,  325 

Plate  levels,  to  adjust,  43 
Plumb  bobs,  35 

Plumbing  bar,  home-made,  152 
Practical  hints — see  the  particular  in- 
strument. 
Precise  level,  adjustments,  249 

tests,  249 

Precise  leveling,  267 
field  routine,  269 
methods,  267 
record,  270 
see  also  Leveling. 
Prismatic  compass,  39 
Probable  error,  defined,  367 
formulas  for  computing,  368 

applications,  372 
examples,  370 
Problems,  373 
Profile,  drawing,  265 
'  leveling— see  Leveling, 
paper,  kinds  of,  266 

Ramsden's  eye-piece,  75 
Refraction,  correction  for,  280 
Rods,  leveling,  227 
Boston,  228 
British,  234 
classified,  227 
Francis,  234 
New  York,  227 
patterns  for  self -reading,  233 
Philadelphia,  228 
self-reading,  232 
patterns  for,  233 
vs.  target,  235 
target,  227 

target  for,  230 
Texas,  234 
stadia,  178 
patterns.  180 
target,  182 


39° 


INDEX. 


BUH— TAP 

Ruhlmann's   method    of    barometric 
leveling,  326 

Smythe's  telemeter,  217 
Spherical  aberration,  80 
Solar  compass,  57 
adjustments,  61 
construction,  58 
defects,  62 
history,  63 
merits,  62 
principle  of,  57 

Solar  transit,  adjustments,  134 
construction,  130 
using,  135 

limits  of  precision,  143 
mine  surveys,  144 
sources  of  error,  140 
Sources  of  error— see  the  instrument 

in  question. 

Stadia,  advantages  of,  208 
c,  to  find,  185 
defined,  173 
/,  to  find,  185 

formula,  fundamental,  182 
horizontal  Hue  of  sight,  182 
inclined  line  of  sight,  191 
horizontal  distance,  192 
vertical  distance,  193 
hairs,  placing  of,  175 
adjustable,  170 
distance  between,  178 
fc,  to  find,  186 
notes,  reducing,  194 
arithmetical  tables,  196 
geometrical  diagrams,  200,  201 
principles.  174 
reducing  the  notes,  194 
arithmetical  tables,  196 
geometrical  diagrams,  200,  201 
rods,  kinds,  178 
graduation,  179 
home-made,  181 
patterns  for,  180 
self-reading,  180 
target  for.  182 

surveying,  limits  of  precision,  204 
practical  hints,  207 
sources  of  error,  202 
inclination  of  rod,  202 
observation,  203 
using— see  Stadia  surveying. 
vs.  gradienter.  214 
Standard,  for  chain,  10 
Surveys,  land,  local  attraction  in,  356 
mine,  local  attraction  in,  349 

Tacheometer,  defined,  173 
Tangent  screw,  96 

abutting,  99 

English,  97 

Gam  bey,  98 

spring,  99 
Tapes,  linen,  9 

metallic,  9 

testing,  12 

standard,  10 

steel,  5 
merits,  8 


TAP— TEA 

Tapes,  steel  reel.  5 
Target,  corner,  for  level  rod,  272 
ordinary,  for  level  rod,  230 

for  stadia  rod,  182 
Telemeter,  Adie's,  217 
Clarke's,  217 
defined,  173 
Eckhold's,  216 
Gautier's,  217 
gradienter,  209 
Smythe's,  217 
stadia,  173 
Struve's,  217 
tacbeometer,  173 
vertical  circle,  215 
Telescope,  aberration  in,  80 
care,  b9 
classified,  72 
construction,  72 
cross-hair  ring,  78 
defining  power,  80 
eye- piece,  74 
erecting,  76 
Huyghen's,  74 
Kellner's.  77 
negative,  74 
positive,  75 
Kamsden's,  75 
flatness  of  field  of,  81 
Galilean,  73 
magnification  of,  84 
measuring.  73 
objective,  73 
parallax.  88 
size  of  field,  82 
slide,  77 

dumpy  level,  240 
transit,  99 
wye  level,  239 
testing,  80 

Tests— see  the  instrument  in  question. 
Texas  leveling  rod,  234 
Transit,  engineer's,  adjustment,  109 
cross  hairs,  109 
eye-piece,  111 
levels,  plate,  109 
telescope,  112 
standards,  111  \ 
vertical  circle,  114 
care,  127 
construction,  91 
graduation,  93 
accuracy, 103 
eccentricity,  104 
lines,  103 

limits  of  precision,  125 
measuring  angles,  117 
repeating,  118 
series,  117 
objective  slide,  99 

test  of.  106 
practical  hints,  115 
sources  of  error,  124 
tests,  1C3 

using,  115— see  also  Transit  sur- 
veying, 
verniers,  94 
vertical  axes,  95 
solar,  adjustments,  134 


INDEX. 


39* 


TEA— VER 

Transit,  solar,  construction,  130 
using,  135 

limits  of  precision,  143 
mine  surveying,  144 
sources  of  error,  140 
surveying,  120 
angle  method,  120 
comparison  of  methods,  128 
quadrant  methods,  120 
traversing,  121 

determining  areas  by,  360 
Tripod,  construction,  31 
setting,  32 

Vernier,  denned,  64 
direct,  65 
least  count,  65 
principle,  64 
reading,  66 
barometer,  67,  69 


VER— WYE 

Vernier  reading  level  rod,  66 

practical  hints,  70 

transit,  67 
retrograde,  65 

Vertical  circles  as  a  gradienter,  215 
Vidi  aneroid,  301 

Weilenmann's    barometric    formula, 

346 

Whitney's  method  of  barometric  level- 
ing, 325 
Williamson's      barometric     formula, 

343 
Williamson's  method   of  barometric 

leveling,  323 

Wire,  to  measure  with,  9 
Wye  level,  adjustments,  241 

collimation,  242 

cross  hairs,  242 

level  tube,  241 

test  level,  246 


AN  INITIAL  PINE  OF  25  CENTS 


TOV  23  1942- 


LD21-100m-7, '40  (6936s) 


YB  40267 


785  8  j 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


